# Hard Geometry Exercise. Need Help !

#### Cacapipi2

In an orthogonal plane space, we consider the hyperbola I of equation xy=a, where a>0
1) Let be M1, M2, M3 three points I which abscissa are respectively x1, x2, x3 pairwise distinct. Determine the equation of the altitude D, coming from M3 ; deduce the coordinates of H, point of intersection of D and I.
2) Show that H is the orthocenter of the triangle M1M2M3.

#### chiro

MHF Helper
Hey Cacapipi2.

First question that has to be asked is - Have you have drawn a diagram and whether you have attempted the question yourself?

#### johng

Surely, you can find the equation of the altitude to be
$$y={x_1x_2\over a}\,(x-x_3)+{a\over x_3}$$
Then all you need do is solve the equation for x:
$${x_1x_2\over a}\,(x-x_3)+{a\over x_3}={a\over x}$$
The above equation (left to you) has solutions
$$x=x_3\text{ and }x={-a^2\over x_1x_2x_3}$$
The second question about the orthocenter should now be obvious.

#### Cacapipi2

I assume I just have to do the same with let's say the point M2 : using this method, I can show that the altitude coming from M2 intersects I on the point H ; but then it intersects also the altitude coming from M3 on the point H. So H must be the orthocenter of the triangle M1M2M3. Thank you very much !

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