Advance Geometry

Question

In unit square ABCD, the inscribed circle w intersects CD at M and AM intersects w at a point P different from M. What is AP?

use the the power of point theorem and the Pythagorean theorem.

Show all the algebra steps

Answer 1

https://imgur.com/a/u4CvXMS

Here is an image of my set-up for the problem. The crucial part of geometry problems described in words is to set up a proper picture. How I created the picture was first drawing a square, and carefully drawing a circle in it. I made sure the circle only touches the square in 4 points so that the circle is properly inscribed in the square.

I then labeled AP as x, and PM as y.

1. Now following the directions of the problem, we apply the Power of Point theorems (https://artofproblemsolving.com/wiki/index.php/Power_of_a_Point_Theorem) to write the first equation in the green box.
2. Then, still following the directions of the problem, we apply the Pythagorean theorem to the right triangle ADM. This yields the second equation in the green box.

Notice that our two equations form a system with variables x and y. So we now use our past algebra knowledge to solve for x.

TIP: Oftentimes, it helps to give short names to unknown lengths; in this case, we said AP was x and PM was y. This allows nicer-looking algebra.