# Cosine and sine rule on a triangle

$$\triangle ABC$$ is isosceles with legs $$AC=BC=b$$ and angle $$\measuredangle ACB=\gamma$$. Let $$CD$$ and $$BM$$ be altitudes that intersect at $$H$$. Find $$MD$$.

I have been struggling with this problem for an hour now. I really don't know how to start. I was thinking about using cosine or sine rule, but it seems useless at the end. For example $$AB^2=AC^2+BC^2-2\cdot AC\cdot BC\cdot \cos\gamma=2b^2-2b^2\cos\gamma$$

• the angle MBA is known, you can get the point M as the intersection of straight lines whose angular coefficients you know. at that point you just need to make the distance between two points of the plane between D and M – Patrick Danzi Jan 21 at 8:46

Hint: Show that quadrilateral MDBC is inscribed. Then, show that $$\angle MAD=\angle AMD$$ so $$\triangle ADM$$ is isosceles with $$MD=AD$$. Finally, you have to express $$AD$$ in terms of $$b$$ and $$\gamma$$.

• Can you give me a hint on how to show $\angle MAD=\angle AMD$? – Katherine Jan 21 at 8:47
• @nicoledobreva $\angle AMD=180^{\circ}-\angle CMD$. Also, your drawing has $\angle ACD=\gamma$ and not $\angle ACB$. That's a mistake, I guess? – bjorn93 Jan 21 at 8:50
• I am trying to find $AB$ using the cosine rule on triangle ABC. I got that $AB=\sqrt{2b^2(1-\cos\gamma)}$. The possible answers are $b\cos\gamma;b\sin\gamma,b\sin\dfrac{\gamma}{2}$ and $b$. Well, using the fact that $MD=AB=\dfrac{1}{2}AB$ we cannot get such answer. – Katherine Jan 21 at 8:59
• I meant $MD=AD=..$ – Katherine Jan 21 at 9:05
• @nicoledobreva There's no contradiction because you haven't simplified this: $2(1-\cos\gamma)=\sin^2(\gamma/2)$. Then take roots. But you can find $AD$ directly from $\triangle ACD$ using the definition of sine. There's really no need to use law of cosines. – bjorn93 Jan 21 at 9:06

$$\therefore MD = AD = b\sin \gamma$$

• A really nice diagram. How did you do it? – Katherine Jan 21 at 9:43
• @nicoledobreva On Geogebra website and exported image to MS Paint. – cosmo5 Jan 21 at 9:45
• I don't see this option for exporting. I have GeoGebra file, PNG image, SVG image, PDF document, 3D Print. Which one do you choose? – Katherine Jan 21 at 9:46
• Oh, I see. And then you exported in MS Paint? Can I ask why? – Katherine Jan 21 at 9:48
• @nicoledobreva To crop the image so it appears big here. You can upvote if you found this answer useful. – cosmo5 Jan 21 at 9:49