Given triangle /_\ABC , with AB=AC and 2 times(length of inradius) which is equal to the length of exradius of excirle opposite vertex A. Setup the coordinate system in the plane of triangle /_\ABC as follow: Origin at vertex B, positive X-axis along BC and positive Y-axis on the A side of BC. Find the equations of the incircle , excircles and circumcircle.

Kale Sampson

Kale Sampson

Answered question

2022-11-05

Given triangle A B C , with A B = A C and 2 times(length of inradius) which is equal to the length of exradius of excirle opposite vertex A. Setup the coordinate system in the plane of triangle A B C as follow: Origin at vertex B, positive X-axis along B C and positive Y-axis on the A side of B C. Find the equations of the incircle , excircles and circumcircle.

Answer & Explanation

Incampo5in

Incampo5in

Beginner2022-11-06Added 10 answers

Hint: Here is a diagram of the construction described above.

There is no absolute scale given in the problem, so lets set the inradius of A B C to be 1. Since the radius of the excircle opposite A is twice that of the incircle, it is 2. By similar triangles, we have | A E | = 2 | A D | . Furthermore, | D E | = 3, so | A D | = 3 and | A E | = 6.
Everything else should be calculable using similar triangles.
Kailyn Hamilton

Kailyn Hamilton

Beginner2022-11-07Added 1 answers

In the coordinate system ( B , B C , B C ) we have B ( 0 , 0 ), C ( 1 , 0 ) and A ( 1 2 , a ) where a = A H : height from vertex A. You have to find a using hypothesis and the rest must be easy.

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