The rhombus and the bisecting diagonal

There is a second condition that turns a parallelogram into a rhombus: it is enough for one diagonal to be the bisector of an angle. The idea is to use parallelism to make an isosceles triangle appear.

Theorem — Rhombus (bisecting diagonal)

If in a parallelogram one diagonal is the bisector of an angle, then the parallelogram is a rhombus.

Proof

The idea is to exploit the parallelism of the sides to turn the bisector hypothesis into an isosceles property.

1. Hypothesis. $ABCD$ a parallelogram. The diagonal $AC$ is the bisector of $\widehat{A}$: $\widehat{DAC}\cong\widehat{BAC}$.
2. Observe. Since $BC\parallel DA$ and $AC$ is a transversal, the alternate interior angles give $\widehat{BCA}\cong\widehat{DAC}$.
3. Deduce. Combining: $\widehat{DCA}\cong\widehat{DAC}$, which makes the triangle $DAC$ isosceles and hence $DC\cong DA$.
4. Verified. Since in a parallelogram the opposite sides are congruent ($DC\cong AB$, $DA\cong BC$), it follows that $DA\cong DC\cong AB\cong BC$: $ABCD$ is a rhombus.

$\blacksquare$

Property — Conditions for the rhombus

A parallelogram is a rhombus if and only if:

1. the diagonals are perpendicular,   or
2. one diagonal is the bisector of an angle.

Links

Topics: Euclidean geometry
Concepts: Bisector · Proof · Parallelogram · Line · Rhombus · Isosceles triangle
Skills: Proving · Synthetic geometry