The bisector of equal angles A and C of an isosceles triangular ABC intersect the lateral sides of the triangle

| August 6, 2021 | education

The bisector of equal angles A and C of an isosceles triangular ABC intersect the lateral sides of the triangle at points E and P, respectively. prove that a four-line to an APEC-trapezoid with three equal sides

Since the triangle ABC is isosceles, the angle BAC = BCA.

Let the angle BAC = 2 * X0, then the angle EAP = BAC / 2 = 2 * X / 2 = X0.

In triangles ACE and ACB, the AC side is common, and the angles about the AC side are equal, then the ACE and ACB triangles are equal in line and two adjacent angles, and therefore AE = CP.

Then BE = BP, and the triangles ABC and BEP are similar in angle B and two adjacent proportional sides, which means AC is parallel to EP, and then the angle EPA = CAP as crosswise, and then the triangle AEP is isosceles, AE = EP.

Then AEPC is a trapezoid for which AE = EP = CP, as required.



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