Through vertex C of triangle ABC there is a line parallel to its bisector AA1 and intersecting line AB

| September 18, 2021 | education

Through vertex C of triangle ABC there is a line parallel to its bisector AA1 and intersecting line AB at point D. Prove that AC = AD.

According to the condition, AA1 is parallel to CD, then the angle ACD is equal to the angle CAA1 as the intersecting angles at the intersection of parallel lines AA1 and CD of the secant AC.

Let the value of the angle CAA1 = X0, then the angle ACD = X0, and the angle BAC = 2 * X0, since AA1 is the bisector of the angle.

The angle BAC and CAD are adjacent angles, the sum of which is 180, then the angle CAD = 180 – 2 * X.

In triangle ACD, the sum of the interior angles will be (180 – 2 * X) + X + ADC = 180.

ADC = X0.

Then the angle АDС = АСD = X0, and the triangle АСD is isosceles and АD = АС, which was required to be proved.



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