# Prove that H is orthocentre of ABC using inversion

Three equal circles pass through a given point $$H$$ and meet one another two by two at $$A,B,C$$ prove that $$H$$ is orthocentre of triangle $$ABC$$.

My try -

I proved it using elementary geometry methods quite easily but i also want to prove it using inversion...

so first i invert about H to make all three circles lines but not able to go further to prove that it is orthocentre of ABC...

Any hints?

Invert at $$H$$. The fact that the three circles through $$H$$ are equal means that, if $$A',B',C'$$ are the images of $$A,B,C$$, then the distances from $$H$$ to $$A'B'$$, $$B'C'$$, and $$C'A'$$ are the same, because the point diametrically opposite $$H$$ on $$(HAB)$$, for example, inverts to the foot from $$H$$ to $$A'B'$$. Therefore $$H$$ is the incenter of $$A'B'C'$$. Now, let $$X,Y,Z$$ be the $$A'$$-, $$B'$$-, and $$C'$$-excenters of $$A'B'C'$$. We see that $$H$$ is the orthocenter of $$XYZ$$ and that a negative inversion about $$H$$ sends $$A'\to X$$, etc., so $$ABC$$ is directly similar at $$H$$ to $$XYZ$$, and so $$H$$ is the orthocenter of $$ABC$$.

• If three circles are equal and passing through H ..how that means that distance from H to A'B' , B'C' ,C'A' are the same.......I think I am missing something ...can you pls provide the missing details.. Mar 25, 2020 at 4:36
• @User12002 I've added an explanation. Mar 25, 2020 at 6:51

Observe an inversion with center at $$C$$ and radius $$r= CH$$.

Let $$A\mapsto A'$$ and $$B\mapsto B'$$.

Since circles (Y) and (Z) have equal radius and go through $$C$$ they map to a lines $$CA'$$ amd $$CB'$$ which are at equal distance from $$C$$. So $$HC$$ is angle bisector for $$\angle A'HB'$$. Now we have $$\angle HBC = \angle B'HC = \angle A'HC = \angle HAC =:\alpha$$

By simmetry we have $$\angle HBA = \angle HCA =:\beta$$ and $$\angle HAB = \angle HCB =:\gamma$$

Now it is clealry $$\alpha +\beta +\gamma = 90^{\circ}$$ so we are done.

• I have 2 doubts in your proof ..1) if radius is equal of both circles then how they map to lines which are equal distance from C ...2) how HBC=B'HC did uu apply angle segment theorem ? Can yu pls give more details to your answer ... Mar 24, 2020 at 3:00