**Basic Geometric Proofs**

__Claim:__

Let A, B, C, D be points on a circle where A is not
equal to B, and C is not equal to D. Suppose that lines AB and CD intersect at
a point P. Then (PA) (PB) = (PC) (PD).

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__Claim:__

If a quadrilateral has a circle inscribe in it, then
the sum of one pair of opposite sides is equal to the sum of the other pair of
opposite sides.

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__Claim:__

For every quadrilateral, the midpoints of its sides
form the vertices of a parallelogram.

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__Claim:__

Claim: If an equilateral polygon is inscribed in a
circle, then it is a regular polygon.

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__Claim__:

Connecting the midpoints of the sides of a triangle
divides the triangle into four little triangles. Show that the little triangles
are similar to the original triangle and that all four of these little
triangles are congruent.

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