p202 q20 re.ex.9
don't know how to do [b][ii]---show that ABC is equilaqteral. thanks
Hello, afeasfaerw23231233!
I've made one observation . . . $\displaystyle \Delta ODE$ is equilateral.
. . But I haven't gone any further.
We have these areas: .$\displaystyle \begin{Bmatrix}\Delta CED & = & \frac{1}{2}ab\cos^2\theta\sin\theta \\ \text{quad }ADEB & = & \frac{1}{2}ab\sin^3\!\theta \end{Bmatrix}$
We are given the ratio of these areas: .$\displaystyle \frac{\Delta CED }{\text{quad }ADEB } \;=\;\frac{\frac{1}{2}ab\cos^2\!\theta\sin\theta}{ \frac{1}{2}ab\sin^3\!\theta} \;=\;\frac{1}{3} $
. . which simplifies to: .$\displaystyle \frac{\cos^2\!\theta}{\sin^2\!\theta} \:=\:\frac{1}{3}\quad\Rightarrow\quad\tan^2\!\thet a \:=\:3\quad\Rightarrow\quad \tan\theta \:=\:\sqrt{3}$
Hence: .$\displaystyle \theta \:=\:60^o$
Since $\displaystyle \Delta ODE$ is isosceles $\displaystyle (OD = OE = \text{radius}),$
. . then: .$\displaystyle \angle DEO = 60^o$ . . . Hence: .$\displaystyle \angle DOE = 60^o$
Therefore: .$\displaystyle \Delta ODE$ is equilateral.