Let △ A B C and E, D on [ A B ] and [...

Banguizb

Banguizb

Answered

2022-06-30

Let A B C and E, D on [ A B ] and [ A C ] s.t. BEDC is inscribable. Let P [ B D ] , Q [ C E ] , s.t. AEPC and ADQB are also inscribable. Show that A P = A Q .

Answer & Explanation

zlepljalz2

zlepljalz2

Expert

2022-07-01Added 22 answers

Step 1
A B D = E B D = E C D = A C E
A B D A C E
A D A B = A E A C
A D A C = A B A E
E P A = E C A = E C D = E B D = P B A
E P A P B A
A E A P = A P A B
A P 2 = A B A E
In the same way: A Q 2 = A C A D
Then A P 2 = A B A E = A C A D = A Q 2 , A P = A Q
rmd1228887e

rmd1228887e

Expert

2022-07-02Added 2 answers

Step 1
A Q R A D P as mentioned in comment.I made two drawings, one isosceles. These points are noticeable:
The measure of the sides and angles of isosceles triangle APQ does not vary with the type of triangle.
It can be assumed as a kind of affine transformation. Because base of triangle is fixed and the location of vertex A alters. It is like moving a triangle APQ relative to the base of ABC to construct three circles with certain measures of diameter.

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