Math

Cyclic and tangential quadrilaterals

Two different tests, two experiments: move D until the opposite angles add to 180° and a circle passes through all four vertices; then build a quadrilateral whose opposite sides have equal sums and a circle touches all four sides. Break each on purpose, and decide which circle is possible.

By Mykhailo Makarov

Actual application image

What happens

A convex quadrilateral can have a circle through its four corners, a circle touching its four sides, both, or neither - and the two are decided by different tests. Through the corners: the opposite angles add up to 180°. Touching the sides: the sums of opposite sides are equal. A rectangle passes the first and fails the second; a kite the other way round.

Equations used in the module

Equations
ABCD convex, vertices in order
a circle through A, B, C, D  ⇔  ∠A + ∠C = ∠B + ∠D = 180°
a circle touching AB, BC, CD, DA  ⇔  AB + CD = BC + DA
an inscribed angle is half the arc it stands on;  the two tangents from a point are equal
the two conditions are independent: a rectangle has the first only, a kite the second only

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