Angle Chasing Helps, What About Dilation?

Geometry Level 2

Three circles of equal radii all intersect at a single point $P$ . Let the other intersections be $A$ , $B$ and $C$ . Which of the following must be true?

P

is the incentre of

\Delta ABC

P

is the orthocenter of

\Delta ABC

P

is the centroid of

\Delta ABC

P

is the circumcentre of

\Delta ABC

1 solution

Ahmad Saad
Apr 25, 2016

Relevant wiki: Triangle Centers - Problem Solving

All the quadrilaterals $PXAZ, PXBY$ and $PZCY$ are rhombus .

$AX$ is equal and parallel to each $ZP$ and $CY$ , then $AXYC$ is a parallelogram,
so $XY$ is equal and parallel to $AC$ .
Since $XY$ and $BP$ are perpendicular then, $BP$ is perpendicular on $AC$ .

Similarly, $ABYZ$ is a parallelogram, so $AB$ is equal and parallel to $ZY$ .
$AB$ is equal and parallel to $ZY$ .
$ZY$ and $CP$ are perpendicular then, $CP$ is perpendicular on $AB$ .

$BXZC$ is a parallelogram.
$XZ$ is equal and parallel to $BC$ .
$XZ$ and $AP$ are perpendicular.
then $AP$ is perpendicular on $BC$ .

Hence, $P$ is the orthocenter of the triangle $ABC$ .

We could also do it this way.AP,CP and BP are the radical axis of the three circles taken pairwise.Since,radical axis is perpendicular to the line joining centres of the two circles, we get AP perpendicular to XZ.But BCZX is a parallelogram as BX is equal and parallel to CZ.Therefore,XZ is parallel to BC.Therefore,AP is perpendicular to BC.Hence,AP is altitude of triangle ABC.Similarly for BP and CP.Hence P is orthocentre of triangle ABC.

Indraneel Mukhopadhyaya - 5 years, 1 month ago

We can also use property of radicle axis(it is perpendicular to line joining center) for the pair of circle.

Dhruv Joshi - 4 years, 2 months ago

See Aha's diagram.Here angle CAP=angle CBP, angle subtended by same chord on congruent circles. Similarly angle PAB=anglePCB and angle PCA =PAB. So PAC+PCA+PBC=90. So AP perpendicular to BC(extended). Similar for CP and BP. Hence, P is the orthocentre.

Prayas Rautray - 3 years, 11 months ago

AP=BP=CP, also AB=BC=CA, so P is circum , in, ortho centre of ABC triangle

Balakrishna Padhy - 3 years, 7 months ago

Angle Chasing Helps, What About Dilation?

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