Finding unknown numbers $x$,$y$ from the given information.

Question

ABC is an arbitrary triangle .Suppose that $\angle BOC$ is $x*\angle (BIC)+y$ degrees. Find $x$ and $y$.

Answer 1

You are given that $\angle BIC=x\times \angle BAC+ y.$ Remember it, we will come back to it.

Now, $\angle BID=\angle IBA+\angle IAB$ and $\angle CID=\angle IAC+ \angle ICA$.

On adding these two you will get

$$\angle BID+\angle CID=\angle IBA+\angle IAB+\angle IAC+ \angle ICA$$ $$\angle BIC=\angle A/2+\angle B/2+\angle A/2+\angle C/2$$ $$\angle BIC=\angle A/2+\angle B/2+\angle C/2+\angle A/2$$

$$\angle BIC=90+\angle A/2$$ $$\angle BIC=90+\angle BAC/2$$ On compairing it to $\angle BIC=x\times \angle BAC+ y.$ tou will get that ${x}=\frac {1}{2} $ and $y=90$