Tagging the pieces of a quadrilateral
A quadrilateral is a shape with four straight sides. Each corner is a vertex. Some quadrilaterals have a circle that touches all four sides.
- Quadrilateral ABCD, with a circle touching all four sides.
- Four touch points, one on each side.
- Equal tangent pieces at each vertex: a, b, c, d.
Four vertices, four equal pairs
A circle inside the quadrilateral that touches all four sides is inscribed. Each vertex is then an outside point of the circle. The two sides at that vertex are tangents from that point.
Two tangents from one outside point are equal. This applies at every vertex. At A the two tangent pieces are equal. Call the shared length a. Likewise b at B, c at C, d at D. Each side is made of two pieces:
AB = a + b, BC = b + c, CD = c + d, DA = d + a.
Add AB and CD: a + b + c + d. Add BC and DA: b + c + d + a. Both sums use the same four letters. So AB + CD = BC + DA. This holds when a circle touches all four sides.
The numbers
Take a = 4, b = 5, c = 6, d = 7. The sides are AB = 9, BC = 11, CD = 13, DA = 11. Check the claim: AB + CD = 9 + 13 = 22. BC + DA = 11 + 11 = 22. They match.
A triangle with a circle inside works the same way. Its tangent pieces are x, y and z, one length per corner. The sides are x + y, y + z and z + x. Each length appears on two sides. So the perimeter is twice the sum of the three pieces.
Suppose the quadrilateral is a parallelogram. Suppose a circle touches all four sides. Then AB = CD and BC = DA. So AB + CD = 2 × AB, and BC + DA = 2 × BC. The rule gives 2 × AB = 2 × BC, so AB = BC. All four sides are equal. So the parallelogram is a rhombus.
Watch out!
The two pieces on one side belong to different corners. They are equal only if a = b. The equal pair is the two pieces at one corner.
A circle touches every side. Tap the missing length.
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