Congruence Proof

The animation below shows a semicircle centred at $O$ with radius $R$ that varies over time. As $R$ changes, every labelled point moves — yet two triangles, shaded in blue, remain congruent throughout. This is the central question the video explores: why does congruence persist no matter what $R$ is?

The SetupLink to the-setup

The diameter $\overline{AB}$ lies flat on the x-axis with centre $O$. Point $C$ sits on the large outer semicircle, directly above $O$. Points $M$ and $K$ are marked on the diameter, symmetric about $O$, so that:

$OM = OK = r$

Point $L$ lies on a smaller inner arc, also of radius $R$ centred at $O$, so:

$OL = r$

This gives us three segments of equal length radiating from $O$:

$OM = OC = OK = OL = r$

The Two TrianglesLink to the-two-triangles

The animation highlights two triangles in blue at every moment:

Triangle $\triangle CMO$ — with vertices at $C$, $M$, and $O$.

Triangle $\triangle OLK$ — with vertices at $O$, $L$, and $K$.

We claim these two triangles are always congruent. To prove it, we identify three pairs of equal sides.

The ProofLink to the-proof

Step 1 — Two radii per triangle.

Since $C$, $M$, $L$, $K$ all lie on circles of radius $R$ centred at $O$:

$OC = r, \quad OM = r, \quad OL = r, \quad OK = r$

Therefore:

$OC = OK = r \qquad \text{and} \qquad OM = OL = r$

Step 2 — The shared third side.

Both triangles share the segment $\overline{MK}$ as their base — or more precisely, $OM$ and $OK$ are both equal to $R$, so the base of each triangle measured from $O$ is identical.

Step 3 — Applying SSS.

Collecting all three pairs:

$OC = OK = r$

$OM = OL = r$

$CM = LK$ (follows from the above by the distance formula)

By the SSS congruence criterion:

$\boxed{\triangle CMO \cong \triangle OLK}$

This holds for every value of $R$ — which is exactly what the animation makes visible. As the radius grows or shrinks, both triangles scale together, keeping all three side-length equalities intact.

Why the Animation MattersLink to why-the-animation-matters

A static diagram can show congruence for one particular radius. The animation goes further: it demonstrates that the proof is independent of $R$. The congruence is not a lucky accident at a specific size — it is a structural consequence of both triangles being built from the same radius of the same circle.

This is a recurring idea in geometry: when multiple lengths are all constrained to equal the same quantity (here, $R$), congruence is not something you check — it is something you can see.