Riemann surface

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My age: 32
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my page - www.hostgator1centcoupon.info Template:Infobox Polygon In geometry, an equilateral triangle is a triangle in which all three sides are equal. In traditional or Euclidean geometry, equilateral triangles are also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. They are regular polygons, and can therefore also be referred to as regular triangles.

Principal properties

Equilateral triangle

Assuming the lengths of the sides of the equilateral triangle are a, we can determine using the Pythagorean theorem that:

• The area is $A={\frac {\sqrt {3}}{4}}a^{2}$
• The perimeter is $p=3a\,\!$
• The radius of the circumscribed circle is $R={\frac {\sqrt {3}}{3}}a$
• The radius of the inscribed circle is $r={\frac {\sqrt {3}}{6}}a$
• The geometric center of the triangle is the center of the circumscribed and inscribed circles
• And the altitude (height) from any side is $h={\frac {\sqrt {3}}{2}}a$.

Many of these quantities have simple relationships to the altitude ("h") of each vertex from the opposite side:

• The area is $A={\frac {h^{2}}{2}}$
• The height of the center from each side is $A={\frac {h}{3}}$
• The radius of the circle circumscribing the three vertices is $R={\frac {2h}{3}}$
• The radius of the inscribed circle is $r={\frac {h}{3}}$

In an equilateral triangle, the altitudes, the angle bisectors, the perpendicular bisectors and the medians to each side coincide.

Derivation of area formula

Using Pythagorean theorem

The area of a triangle is half the base times the height.

$A={\frac {1}{2}}bh$

The height of an equilateral triangle can be found using Pythagorean theorem.

An equilateral triangle with a side of 2 has a height of Template:Sqrt and the sine of 60° is Template:Sqrt/2.

The legs of the right triangle are half of the base and the height, while the hypotenuse is the side or base of the equilateral triangle.

$({\frac {b}{2}})^{2}+h^{2}=b^{2}$

The formula for the height can be derived by rearranging the equation and finding the square root.

$h={\sqrt {b^{2}-({\frac {b}{2}})^{2}}}={\sqrt {b^{2}-{\frac {b^{2}}{4}}}}={\sqrt {{\frac {3}{4}}b^{2}}}={\frac {\sqrt {3}}{2}}b$

By substituting $h={{\tfrac {\sqrt {3}}{2}}b}$, the formula for the area of an equilateral triangle can be derived.

$A={\frac {1}{2}}b\times {\frac {\sqrt {3}}{2}}b={\frac {\sqrt {3}}{4}}b^{2}$

Using trigonometry

Using trigonometry, the area of a triangle with any two sides a and b, and an angle C between them is:

$A={\frac {1}{2}}ab\sin C$

Each angle of an equilateral triangle is 60°.

$A={\frac {1}{2}}ab\sin 60^{\circ }$

The sine of 60° is ${\tfrac {\sqrt {3}}{2}}$.

$A={\frac {1}{2}}ab\times {\frac {\sqrt {3}}{2}}={\frac {\sqrt {3}}{4}}ab$

All sides of an equilateral triangle are equal, therefore a = b.

$A={\frac {\sqrt {3}}{4}}a^{2}$

Characterizations

A triangle ABC that has the sides a, b, c, semiperimeter s, area T, exradii r_a, r_b, r_c (tangent to a, b, c respectively), and where R and r are the radii of the circumcircle and incircle respectively, is equilateral if and only if any one of the statements in the following eight categories is true. These are also properties of an equilateral triangle.

Sides

• $\displaystyle a^{2}+b^{2}+c^{2}=ab+bc+ca.$
• $\displaystyle abc=(a+b-c)(a-b+c)(-a+b+c)\quad {\text{(Lehmus)}}$
• $\displaystyle {\frac {1}{a}}+{\frac {1}{b}}+{\frac {1}{c}}={\frac {\sqrt {25Rr-2r^{2}}}{4Rr}}.$

Semiperimeter

• $\displaystyle s=2R+(3{\sqrt {3}}-4)r\quad {\text{(Blundon)}}$
• $\displaystyle s^{2}=3r^{2}+12Rr.$
• $\displaystyle s^{2}=3{\sqrt {3}}T.$
• $\displaystyle s=3{\sqrt {3}}r$
• $\displaystyle s={\frac {3{\sqrt {3}}}{2}}R$

Angles

• $\displaystyle A=B=C=60^{\circ }$
• $\displaystyle \cos {A}+\cos {B}+\cos {C}={\frac {3}{2}}$
• $\displaystyle \sin {\frac {A}{2}}\sin {\frac {B}{2}}\sin {\frac {C}{2}}={\frac {1}{8}}.$

Area

• $\displaystyle T={\frac {a^{2}+b^{2}+c^{2}}{4{\sqrt {3}}}}\quad {\text{(Weizenbock)}}$
• $\displaystyle T={\frac {\sqrt {3}}{4}}(abc)^{^{\frac {2}{3}}}.$

Circumradius, inradius and exradii

• $\displaystyle R=2r\quad {\text{(Chapple-Euler)}}$
• $\displaystyle 9R^{2}=a^{2}+b^{2}+c^{2}.$
• $\displaystyle r={\frac {r_{a}+r_{b}+r_{c}}{9}}.$
• $\displaystyle r_{a}=r_{b}=r_{c}.$

Equal cevians

Three kinds of cevians are equal for (and only for) equilateral triangles:

• The three altitudes have equal lengths.
• The three medians have equal lengths.
• The three angle bisectors have equal lengths.

Coincident triangle centers

Every triangle center of an equilateral triangle coincides with its centroid, and for some pairs of triangle centers, the fact that they coincide is enough to ensure that the triangle is equilateral. In particular, a triangle is equilateral if any two of the circumcenter, incenter, centroid, or orthocenter coincide. It is also equilateral if its circumcenter coincides with the Nagel point, or if its incenter coincides with its nine-point center.

Six triangles formed by partitioning by the medians

For any triangle, the three medians partition the triangle into six smaller triangles.

• A triangle is equilateral if and only if any three of the smaller triangles have either the same perimeter or the same inradius.Primarily based on the most recent URA personal property value index (PPPI) flash estimates, we know that the PPPI, which represents the overall real property price development, has dipped in 2013Q4. That is the first dip the market has seen within the final 2 years.
  
  To give you some perspective, the entire number of personal properties in Singapore (together with govt condominiums) is 297,689 in 2013Q3. Primarily based on the projection that there will be 19,302 units accomplished in 2014, the rise in residential models works out to be more than 6%. With a lot New Ec Launch Singapore provide, buyers might be spoilt for alternative and this in flip will lead to their reluctance to pay a premium for potential models. The complete textual content of the Copyright Act (Cap sixty three) and different statutes regarding IPR might be found on the Singapore Statutes Online Website online The Group's income jumped forty.1 p.c to $324.5 million from $231.6 million in FY 2013, lifted by increased development income and sales of growth properties in Singapore and China. Actual Estate Shopping for
  
  One factor we've on this nation is a big group of "economists," and "market analysts." What's interesting about this group of real property market-watchers is that there are two very other ways wherein they predict Boomers will affect housing markets over the subsequent decade. Let's check out those two opposites and see how every can change the best way real property investors strategy their markets. The good news is that actual property buyers are prepared for either state of affairs, and there's profit in being ready. I'm excited and searching ahead to the alternatives both or each of these conditions will supply; thank you Boomers! Mapletree to further broaden past Asia Why fortune will favour the brave in Asia's closing real property frontier
  
  The story of the 23.2 home begins with a stack of Douglas fir beams salvaged from varied demolished warehouses owned by the consumer's household for a number of generations. Design and structure innovator Omer Arbel, configured them to type a triangulated roof, which makes up one of the placing features of the home. The transient from the entrepreneur-proprietor was not solely to design a house that integrates antique wood beams, however one which erases the excellence between inside and exterior. Built on a gentle slope on a large rural acreage surrounded by two masses of previous-development forests, the indoors movement seamlessly to the outdoors and, from the within looking, one enjoys unobstructed views of the existing natural panorama which is preserved
  
  First, there are typically extra rental transactions than gross sales transactions, to permit AV to be decided for each property primarily based on comparable properties. Second, movements in sale costs are more unstable than leases. Hence, utilizing rental transactions to derive the AV helps to maintain property tax more steady for property homeowners. If you're buying or trying to lease a property. It's tiring to call up individual property agent, organize appointments, coordinate timing and to go for particular person property viewing. What most individuals do is to have a property agent representing them who will arrange and coordinate the viewings for all the properties out there based mostly on your necessities & most well-liked timing. Rent Property District 12 Rent Property District thirteen
  
  The Annual Worth of a property is mostly derived based mostly on the estimated annual hire that it may well fetch if it have been rented out. In determining the Annual Worth of a property, IRAS will think about the leases of similar properties within the vicinity, dimension and condition of the property, and different relevant components. The Annual Worth of a property is determined in the identical method regardless of whether the property is let-out, proprietor-occupied or vacant. The Annual Worth of land is determined at 5% of the market price of the land. When a constructing is demolished, the Annual Worth of the land is assessed by this method. Property Tax on Residential Properties Buyer Stamp Responsibility on Buy of Properties – Business and residential properties Rent House District 01
  
  Within the event the Bank's valuation is decrease than the acquisition price, the purchaser has to pay the distinction between the purchase value and the Bank's valuation utilizing money. As such, the money required up-front might be increased so it's at all times essential to know the valuation of the property before making any offer. Appoint Lawyer The Bank will prepare for a proper valuation of the property by way of physical inspection The completion statement will present you the balance of the acquisition price that you must pay after deducting any deposit, pro-rated property tax and utility costs, upkeep prices, and different relevant expenses in addition to any fees payable to the agent and the lawyer. Stamp Responsibility Primarily based on the Purchase Price or Market Value, whichever is larger

• A triangle is equilateral if and only if the circumcenters of any three of the smaller triangles have the same distance from the centroid.Primarily based on the most recent URA personal property value index (PPPI) flash estimates, we know that the PPPI, which represents the overall real property price development, has dipped in 2013Q4. That is the first dip the market has seen within the final 2 years.
  
  To give you some perspective, the entire number of personal properties in Singapore (together with govt condominiums) is 297,689 in 2013Q3. Primarily based on the projection that there will be 19,302 units accomplished in 2014, the rise in residential models works out to be more than 6%. With a lot New Ec Launch Singapore provide, buyers might be spoilt for alternative and this in flip will lead to their reluctance to pay a premium for potential models. The complete textual content of the Copyright Act (Cap sixty three) and different statutes regarding IPR might be found on the Singapore Statutes Online Website online The Group's income jumped forty.1 p.c to $324.5 million from $231.6 million in FY 2013, lifted by increased development income and sales of growth properties in Singapore and China. Actual Estate Shopping for
  
  One factor we've on this nation is a big group of "economists," and "market analysts." What's interesting about this group of real property market-watchers is that there are two very other ways wherein they predict Boomers will affect housing markets over the subsequent decade. Let's check out those two opposites and see how every can change the best way real property investors strategy their markets. The good news is that actual property buyers are prepared for either state of affairs, and there's profit in being ready. I'm excited and searching ahead to the alternatives both or each of these conditions will supply; thank you Boomers! Mapletree to further broaden past Asia Why fortune will favour the brave in Asia's closing real property frontier
  
  The story of the 23.2 home begins with a stack of Douglas fir beams salvaged from varied demolished warehouses owned by the consumer's household for a number of generations. Design and structure innovator Omer Arbel, configured them to type a triangulated roof, which makes up one of the placing features of the home. The transient from the entrepreneur-proprietor was not solely to design a house that integrates antique wood beams, however one which erases the excellence between inside and exterior. Built on a gentle slope on a large rural acreage surrounded by two masses of previous-development forests, the indoors movement seamlessly to the outdoors and, from the within looking, one enjoys unobstructed views of the existing natural panorama which is preserved
  
  First, there are typically extra rental transactions than gross sales transactions, to permit AV to be decided for each property primarily based on comparable properties. Second, movements in sale costs are more unstable than leases. Hence, utilizing rental transactions to derive the AV helps to maintain property tax more steady for property homeowners. If you're buying or trying to lease a property. It's tiring to call up individual property agent, organize appointments, coordinate timing and to go for particular person property viewing. What most individuals do is to have a property agent representing them who will arrange and coordinate the viewings for all the properties out there based mostly on your necessities & most well-liked timing. Rent Property District 12 Rent Property District thirteen
  
  The Annual Worth of a property is mostly derived based mostly on the estimated annual hire that it may well fetch if it have been rented out. In determining the Annual Worth of a property, IRAS will think about the leases of similar properties within the vicinity, dimension and condition of the property, and different relevant components. The Annual Worth of a property is determined in the identical method regardless of whether the property is let-out, proprietor-occupied or vacant. The Annual Worth of land is determined at 5% of the market price of the land. When a constructing is demolished, the Annual Worth of the land is assessed by this method. Property Tax on Residential Properties Buyer Stamp Responsibility on Buy of Properties – Business and residential properties Rent House District 01
  
  Within the event the Bank's valuation is decrease than the acquisition price, the purchaser has to pay the distinction between the purchase value and the Bank's valuation utilizing money. As such, the money required up-front might be increased so it's at all times essential to know the valuation of the property before making any offer. Appoint Lawyer The Bank will prepare for a proper valuation of the property by way of physical inspection The completion statement will present you the balance of the acquisition price that you must pay after deducting any deposit, pro-rated property tax and utility costs, upkeep prices, and different relevant expenses in addition to any fees payable to the agent and the lawyer. Stamp Responsibility Primarily based on the Purchase Price or Market Value, whichever is larger

Famous theorems

Morley's trisector theorem states that, in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle.

Napoleon's theorem states that, if equilateral triangles are constructed on the sides of any triangle, either all outward, or all inward, the centers of those equilateral triangles themselves form an equilateral triangle.

A version of the isoperimetric inequality for triangles states that the triangle of greatest area among all those with a given perimeter is equilateral.

Viviani's theorem states that, for any interior point P in an equilateral triangle, with distances d, e, and f from the sides, d + e + f = the altitude of the triangle, independent of the location of P.

Pompeiu's theorem states that, if P is an arbitrary point in an equilateral triangle ABC, then there exists a triangle with sides of length PA, PB, and PC.

Other properties

By Euler's inequality, the equilateral triangle has the smallest ratio R/r of the circumradius to the inradius of any triangle: specifically, R/r = 2.

The triangle of largest area of all those inscribed in a given circle is equilateral; and the triangle of smallest area of all those circumscribed around a given circle is equilateral.

The ratio of the area of the incircle to the area of an equilateral triangle, ${\frac {\pi }{3{\sqrt {3}}}}$, is larger than that of any non-equilateral triangle.

The ratio of the area to the square of the perimeter of an equilateral triangle, ${\frac {1}{12{\sqrt {3}}}},$ is larger than that for any other triangle.

Given a point in the interior of an equilateral triangle, the ratio of the sum of its distances from the vertices to the sum of its distances from the sides equals 2 and is less than that of any other triangle. This is the Erdős–Mordell inequality; a stronger variant of it is Barrow's inequality, which replaces the perpendicular distances to the sides with the distances to the points where the angle bisectors cross the sides.

For any point P in the plane, with distances p, q, and t from the vertices A, B, and C respectively,

$\displaystyle 3(p^{4}+q^{4}+t^{4}+a^{4})=(p^{2}+q^{2}+t^{2}+a^{2})^{2}.$

For any point P on the inscribed circle of an equilateral triangle, with distances p, q, and t from the vertices,

$\displaystyle 4(p^{2}+q^{2}+t^{2})=5a^{2}$

and

$\displaystyle 16(p^{4}+q^{4}+t^{4})=11a^{4}.$

For any point P on the minor arc BC of the circumcircle, with distances p, q, and t from A, B, and C respectively,

$\displaystyle p=q+t$

and

$\displaystyle q^{2}+qt+t^{2}=a^{2};$

moreover, if point D on side BC divides PA into segments PD and DA with DA having length z and PD having length y, then

$z={\frac {t^{2}+tq+q^{2}}{t+q}},$

which also equals ${\tfrac {t^{3}-q^{3}}{t^{2}-q^{2}}}$ if t ≠ q; and

${\frac {1}{q}}+{\frac {1}{t}}={\frac {1}{y}}.$

An equilateral triangle is the most symmetrical triangle, having 3 lines of reflection and rotational symmetry of order 3 about its center. Its symmetry group is the dihedral group of order 6 D₃.

Equilateral triangles are the only triangles whose Steiner inellipse is a circle (specifically, it is the incircle).

A regular tetrahedron is made of four equilateral triangles.

Equilateral triangles are found in many other geometric constructs. The intersection of circles whose centers are a radius width apart is a pair of equilateral arches, each of which can be inscribed with an equilateral triangle. They form faces of regular and uniform polyhedra. Three of the five Platonic solids are composed of equilateral triangles. In particular, the regular tetrahedron has four equilateral triangles for faces and can be considered the three dimensional analogue of the shape. The plane can be tiled using equilateral triangles giving the triangular tiling.

Geometric construction

Construction of equilateral triangle with compass and straightedge

An equilateral triangle is easily constructed using a compass and straightedge. Draw a straight line, and place the point of the compass on one end of the line, and swing an arc from that point to the other point of the line segment. Repeat with the other side of the line. Finally, connect the point where the two arcs intersect with each end of the line segment

Alternate method:

Draw a circle with radius r, place the point of the compass on the circle and draw another circle with the same radius. The two circles will intersect in two points. An equilateral triangle can be constructed by taking the two centers of the circles and either of the points of intersection.

The proof that the resulting figure is an equilateral triangle is the first proposition in Book I of Euclid's Elements.

In culture and society

Equilateral triangles have frequently appeared in man made constructions:

• Some archaeological sites have equilateral triangles as part of their construction, for example Lepenski Vir in Serbia.

See also

• Almost-equilateral Heronian triangle
• Dragon's Eye (symbol)
• Isosceles triangle
• Isoperimetric inequality
• Right triangle
• Trigonometry

External links

• 22 year-old Systems Analyst Rave from Merrickville-Wolford, has lots of hobbies and interests including quick cars, property developers in singapore and baking. Always loves visiting spots like Historic Monuments Zone of Querétaro.
  
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