Equilateral Triangle: An equilateral triangle has three congruent sides and three congruent angles. Angles Theorem Corollary to the Base Angles If a triangle is equilateral, then it is equiangular. Given triangle ABC with side lengths a,b,c. 3 9-lines Theorem Consider three nested ellipses and 9 lines tangent to the innermost one. the following theorem. Corollary 4-2 - Each angle of an equilateral triangle measures 60. And you actually know what that measure is. 7:15. Related material Triangle Inequality Theorem Converse. Consider Napoleon's triangles $ABC',\,$ $BCA',\,$ $CAB'.\,$ The Fermat-Torricelli point $F\,$ is the intersection of $AA',\,$ $BB',\,$ $CC'.\,$ It is also a common point of the three circumcircles $(ABC'),\,$ $(BCA'),\,$ $(CAB')\,$ whose centers we denote $C_0,\,$ $A_0,\,$ and $B_0,\,$ respectively. In geometry, an equilateral triangle is a triangle in which all three sides have the same length. Finally, connect the point where the two arcs intersect with each end of the line segment. Converse of Thales Theorem If two sides of a triangle are divided in the same ratio by a line then the line must be parallel to the third side. So, if all three sides of the triangle are congruent, then all of the angles are congruent or each. An equilateral triangle can be constructed by taking the two centers of the circles and either of the points of intersection. All that remains is to expand the diagram by a factor $\sqrt{3}.$, We choose an arbitrary point $M\,$ and construct points $A_1,B_1,C_1\,$ such that $A_1M=a,\,$ $B_1M=b\,$ $C_1M=c\,$ and $\angle B_1MC_1=\angle A+\displaystyle\frac{\pi}{3},\,$ $\angle C_1MA_1=\angle B+\displaystyle\frac{\pi}{3},\,$ $\angle A_1MB_1=\angle C+\displaystyle\frac{\pi}{3}.\,$ It is easily verifies that $\angle B_1MC_1+\angle C_1MA_1+\angle A_1MB_1=2\pi.$, Now compute the side length of $\Delta A_1B_1C_1:$, $\displaystyle\begin{align} − In the figure above, drag both loose ends down on to the line segment C, to see why this is so. Theorem 4-13 Converse of the Isosceles Triangle Theorem If a triangle has two congruent angles, then the triangle is isosceles and the congruent sides are opposite the congruent angles. If the original conditional statement is false, then the converse will also be false. (9x – 11) cm Corollary to the Converse of the Base Angles Theorem: If a triangle is equiangular, then it is equilateral. If a triangle has two congruent sides, does the triangle also have two congruent angles? equiangular. of 1 the triangle is equilateral if and only if[17]:Lemma 2. It is also a regular polygon, so it is also referred to as a regular triangle. If P is on the circumcircle then the sum of the two smaller ones equals the longest and the triangle has degenerated into a line, this case is known as Van Schooten's theorem. A version of the isoperimetric inequality for triangles states that the triangle of greatest area among all those with a given perimeter is equilateral.[12]. Also, the three angles of the equilateral triangle are congruent and equal to 60 degrees. 1 Given a point P 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 is greater than or equal to 2, equality holding when P is the centroid. Construction 2 is by Chris van Tienhoven. White Boards: If