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Let $ABC$ be a triangle and $\Gamma$ its circumcircle. On sides $AC$, $BC$ lies respectively points $E$, $F$ such that $CE=BE$ and $CF=AF$. $CM$ is a median of triangle $EFC$. Show that line $CM$ pass throught point $K$ which is point where meets tangents to circle $\Gamma$ at points $A, B$.

Here is my sketch, I know that $E, F, A, B$ lies on a common circle because triangle $EFC$ and $ABC$ are similar, but don't know what to do now.

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  • $\begingroup$ Your diagram has the line CM going through a 4th point, the intersection of EB and AF. $\endgroup$
    – zyx
    Oct 13 '13 at 3:14
  • $\begingroup$ @zyx: Of course, you should never trust a diagram. In this case, the figure is misleadingly symmetric. If you look at (and decide to trust) the figures accompanying my answer, you'll see that $\overline{CM}$ doesn't necessarily concur with $\overline{EB}$ and $\overline{AF}$. (I broke the symmetry on purpose.) $\endgroup$
    – Blue
    Oct 13 '13 at 3:21
  • $\begingroup$ @Blue, thanks, that answers the implied question. $\endgroup$
    – zyx
    Oct 13 '13 at 3:23
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Because $\triangle AFC$ and $\triangle BEC$ are isosceles, we have $$\angle C \cong \angle CAF \cong CBE$$

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(Note that $\angle C$ is necessarily acute.) Moreover, the bisectors of $\angle AEB$ and $\angle BFA$ create four more copies of $\angle C$. That two of these copies are $\angle HEA$ and $\angle HFB$ implies that $\square HEFC$ is a parallelogram, the diagonals of which bisect each other. Thus, $\overline{CH}$ passes through $M$.

enter image description here

It remains to show that $H$ coincides with $K$.

Note that $\angle EHF$ is yet another copy of $\angle C$ (as the opposite angle in the parallelogram). Since $\angle EAF$ and $\angle EBF$ are too, the points $H$, $A$, $B$ are concyclic with $E$ and $F$ (because they subtend the same angle with chord $\overline{EF}$).

enter image description here

This implies that $\angle ABH$ (which must be congruent to $\angle AFH$, as both subtend chord $\overline{AH}$) is also a copy of $\angle C$; likewise, $\angle BAH \cong \angle C$.

enter image description here

We see, then, that $H$ is the apex of the[*] isosceles triangle with base segment $\overline{AB}$ and base angles congruent to $\angle C$. This description also fits $K$ (why?), so the points must coincide.

enter image description here


[*] There are, of course, two such isosceles triangles with base $\overline{AB}$. "The" triangle in question is the one on the opposite side of $\overleftrightarrow{AB}$ from $C$. (Note that $\angle C$'s acuteness is key to making this distinction.)

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(These are complete steps, you just need to fill in the minor details.)

Step 1: Show that $AEFBK$ is con cyclic, as drawn in your diagram. It could be easier to show that $O$, the center of $\Gamma$, also lies on this circle.

Step 2: Show that $EF=KB=KA$, because they each subtend the same angle in this circle.

Step 3: Show that $KE \parallel BF$ and $KF \parallel EA$.

Step 4: Hence show that $KECF$ is a parallelogram.

Hence $CK$ bisects $EF$, so $CKM$ is a straight line.

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