Unizor - Geometry2D - Circles - Mini Theorem 4
Channel: Zor Shekhtman on UNIZOR Education
Date: 2012-01-13

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Theorem 1
Sum of opposite angles in a convex quadrangle inscribed into a circle is equal to 180 degrees.
Theorem 2 (inverse to Theorem 1)
If sum of opposite angles of a convex quadrangle is equal to 180 degrees, it can be inscribed into a circle.
Theorem 3 (immediate consequence of theorem 1)
(a) Among all parallelograms, only a rectangle can be inscribed into a circle.
(b) Among all trapezoids, only the one with congruent and non-parallel sides (i.e. not a parallelogram) can be inscribed into a circle.
Theorem 4
A quadrangle that circumscribes a circle (i.e. all four its sides are tangent to a circle) has sums of opposite sides equal to each other.
Theorem 5 (four special points in a triangle)
(a) Three perpendicular side bisectors intersect in one point (center of circumscribed circle).
(b) Three angle bisectors intersect in one point (center of inscribed circle).
(c) Three altitudes intersect in one point.
(d) Three medians intersect in one point (center of gravity).
Theorem 6
Given two points on a plane K and L. Consider all the straight lines that contain point K and perpendiculars to these lines from point L. Points where these perpendiculars meet the corresponding lines form a circle with a segment KL as a diameter.
Theorem 7
Given a circle with a center L and a point K inside it. Consider all chords that contain this point K. Midpoints of these chords lie on a circle with segment KL as a diameter.
Theorem 8
Given two circles tangent to each other. Any secant that contains a point of tangency forms two chords in these circles. Prove that two central angles supported by arcs that correspond to two chords are congruent to each other.