One diagonal, two triangles
A diagonal joins two opposite corners. Convex — a shape with no corner pushed in. A diagonal of a convex four-sided shape cuts it into two triangles.
- Quadrilateral ABCD, with diagonal AC drawn.
- A lands exactly on C, and B lands exactly on D.
- Angles BAC and DCA match. They force AB parallel to DC.
- Angles BCA and DAC match too. They force BC parallel to AD.
The rule
Name the corners A, B, C, D in order round the shape. Take ABCD convex. Then B and D lie on opposite sides of AC.
Claim: Suppose triangle ABC fits onto triangle CDA. Let A go onto C, B onto D. In that case ABCD is a parallelogram. "Lands on" means fits exactly on top.
Angle BAC lands on angle DCA, so the two angles are equal. They lie on opposite sides of AC. AC crosses the lines AB and DC. That makes them an alternate pair. Equal alternate angles force AB parallel to DC.
Angles BCA and DAC are the other alternate pair. They force BC parallel to AD. Both pairs are parallel. So ABCD is a parallelogram.
| Step | What happens | Reason |
|---|---|---|
| Cut | AC splits convex ABCD into two triangles | Every diagonal of a convex 4-gon does this |
| Land | Triangle ABC lands on triangle CDA, with A on C and B on D | The claim assumes it |
| Match | Angle BAC equals angle DCA | Matching parts of congruent triangles |
| Force | AB is parallel to DC | Equal alternate angles |
The landing is the whole argument. Most four-sided shapes fail it.
One diagonal of a kite also cuts two matching triangles. A kite has two pairs of equal neighbouring sides. But these triangles match by a fold. A goes onto A, not onto C. No alternate pair appears, and no name is forced.
Turn triangle ABC. Does B land on D?
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