I have a circle of radius $r$ and a triangle inside that circle. Specifically, if you have the triangle $\triangle ABC$ inside a circle with only the one side $AB$ and an angle $\angle \text{B}$ opposite to the other side $AC$ known along with circle radius $r$, could you determine its area?
[Math] Area of a Triangle Inside a Circle
areacirclesgeometrytriangles
Related Solutions
Well, since the distances form a Pythagorean triple the choice was not that random. You are on the right track and reflection is a great idea, but you need to take it a step further.
Check that in the (imperfect) drawing below $\triangle RBM$, $\triangle AMQ$, $\triangle MPC$ are equilateral, since they each have two equal sides enclosing angles of $\frac{\pi}{3}$. Furthermore, $S_{\triangle ARM}=S_{\triangle QMC}=S_{\triangle MBP}$ each having sides of length 3,4,5 respectively (sometimes known as the Egyptian triangle as the ancient Egyptians are said to have known the method of constructing a right angle by marking 12 equal segments on the rope and tying it on the poles to form a triangle; all this long before the Pythagoras' theorem was conceived)
By construction the area of the entire polygon $ARBPCQ$ is $2S_{\triangle ABC}$
On the other hand
$$ARBPCQ= S_{\triangle AMQ}+S_{\triangle MPC}+S_{\triangle RBM}+3S_{\triangle ARM}\\=\frac{3^2\sqrt{3}}{4}+\frac{4^2\sqrt{3}}{4}+\frac{5^2\sqrt{3}}{4}+3\frac{1}{2}\cdot 3\cdot 4 = 18+\frac{25}{2}\sqrt{3}$$
Hence
$$S_{\triangle ABC}= 9+\frac{25\sqrt{3}}{4}$$
The angle $(\angle BAQ)$ is a tangent (chord) angle.
Theorem An Angle formed by a chord and a tangent that intersect on a circle is half the measure of the intercepted arc. So $$\theta=\frac{1}{2} m \overset{\frown}{(AB)}$$ from there $(\angle AOB)=2\theta$
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Best Answer
The angle at $C$ must be $\arcsin\frac{|AB|}{2r}$, namely half of the arc spanned by $AB$. This gives you all the angles in the triangle.
The law of sines then gives you the sides.
And then there are many ways to get the area, such as Heron's formula, or the sine of one angle times half the product of the adjacent sides.