This is a Geometry theorem based on Angles adding up to 180 degrees. It is helpful for Mathematics Olympiad. Try to prove the statement!

**Statement: Angles adding up to 180 degrees**

ABC be an isosceles triangle with AB = AC. P be a point inside the triangle such that, . Suppose M is the midpoint of BC. Show that

**Discussion:**

Our first claim is, AB and AC are tangents to the circum circle of BPC (prove this). Also extend AP to meet the circum circle at G again. It is sufficient to show .

Next we claim that IPCO and MPGO are cyclic (how?) .

Let

- as MPGO is cyclic
- as IPCO is cyclic

So

Also as OP = OC (radii), hence = =

Hence done.

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