Radius of the circumscribed circle of an isosceles triangle

Question

An isosceles triangle $ABC$ is given $(AC=BC).$ The perimeter of $\triangle ABC$ is $2p$, and the base angle is $\alpha.$ Find the radius of the circumscribed circle $R$.

$$R=\frac{p}{2\sin\alpha(1+\cos\alpha)}$$

Let $CD=2R.$ The triangle $BCD$ is a right triangle and we have $\angle BAC=\angle ABC=\angle BDC=\alpha.$

I am not sure how to approach the problem. It's really hard for me to solve problems like this. Can you give me a hint and some thoughts on the problem?

Answer 1

Another simple approach. Let $x=AC=BC$. Then

$$2p=AC+BC+2AH\\=2x+2x\cos\alpha$$

and

$$R=\frac 12 CD=\frac 12 \frac{BC}{ \sin \alpha} = \frac{x}{2 \sin \alpha}$$

Now you can complete the solution by a simple substitution.

Answer 2

The diameter is $$CD = 2R = \sqrt{BD^2 + BC^2}$$ by the Pythagorean theorem, since $\angle CBD$ is inscribed in a semicircle, thus is a right angle.

Now use the trigonometric properties to deduce that $$BH = BD \sin \alpha,$$ and $$ BH = BC \cos \alpha.$$ We also have $$BH + BC = p,$$ because this is half the perimeter of $\triangle ABC$. Now all that is left is to eliminate $BH$, $BD$, and $BC$ from these four equations.

Answer 3

Hint: Use following formula:

$$R=\frac{p}{4\cos\left(\frac{\alpha}{2}\right)\cos\left(\frac{\beta}{2}\right)\cos\left(\frac{\gamma}{2}\right)}$$

Where $\alpha$, $\beta$ and $\gamma$ are angles on vertices A, B and C respectively. $\alpha=\beta$ , therefore we have:

$$R=\frac{p}{4\cos^2\left(\frac{\alpha}{2}\right)\cos\left(\frac{\gamma}{2}\right)}$$

And also:

$$2\alpha+\gamma=\pi$$

$$\implies\frac{\gamma}{2}=\frac{\pi}{2}-\frac{\alpha}{4}$$

Finally:

$$R=\frac{p}{4\cos^2\left(\frac{\alpha}{2}\right)\sin\left(\frac{\alpha}{4}\right)}$$

Answer 4

Using two known general expressions for the area of $\triangle ABC$

\begin{align} S&=\rho r \tag{1}\label{1} ,\\ S& =2\,R^2\sin\alpha\sin\beta\sin\gamma =4\,R^2\sin^3\alpha\cos\alpha \tag{2}\label{2} , \end{align}

and the expression for the inradius of $\triangle ABC$ in terms of its semiperimeter $\rho$,

\begin{align} r&= \rho\tan\tfrac\alpha2\tan\tfrac\beta2\tan\tfrac\gamma2 = \rho\tan^2\tfrac\alpha2\cot\alpha \tag{3}\label{3} , \end{align} we can find that

\begin{align} R&= \frac1{2\sin\alpha}\,\sqrt{ \frac{\rho r}{\sin\alpha\cos\alpha} } \tag{4}\label{4} ,\\ R&= \frac1{2\sin\alpha}\,\sqrt{ \frac{\rho^2 \tan^2\tfrac\alpha2\cot\alpha}{\sin\alpha\cos\alpha} } =\dots= \frac\rho{2\sin\alpha+\sin2\alpha} \tag{5}\label{5} . \end{align}