Given any non-isosceles triangle $\triangle ABC$, and denoting $AB$ its longest side, the following construction enter image description here

determines the points $DFGE$. In this post is shown that the points $DFCGE$ always determine a circle.

Let then consider the center $J$ of this circle: enter image description here

By means of this new point $J$, we can draw the three circles passing by $F,C$ and $J$, by $C,G$ and $J$, and by $D,E$ and $J$.

The two intersections of the first two circles with the third one determine two additional points $H$ and $I$.

enter image description here

A fourth circle passing by these two points $H$, $I$, and $C$ defines the two points $K$ and $L$ on the sides $AC$ and $BC$, respectively. enter image description here

My conjecture is that the triangle $\triangle LKC$ is similar to $\triangle ABC$.

enter image description here

I tried to apply the techniques suggested in this post and in this other post to prove the claim, but with no success.

Thanks for your suggestions!

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    $\begingroup$ congruent or similar? $\endgroup$ – Vasya Aug 3 '18 at 16:47
  • $\begingroup$ Similar, sorry! I edit immediately! $\endgroup$ – user559615 Aug 3 '18 at 16:49
  • $\begingroup$ 1) If C of $\triangle KCL$ is mapped onto B of $\triangle ABC$, then $\triangle ABC$ must be isosceles in the first place, 2) Just draw the picture. If your conjecture is correct, the sharpest angle (= $\gamma$) should be of the same size as (= $\delta$). $\endgroup$ – Mick Aug 3 '18 at 17:26
  • $\begingroup$ @Mick Maybe you mean that I should write $\triangle ACB$ similar to $\triangle LCK$ $\endgroup$ – user559615 Aug 3 '18 at 17:31
  • $\begingroup$ @greedoid OK! I will try! $\endgroup$ – user559615 Aug 3 '18 at 17:32

Let us at first show that a)$H$ is the intersection of $FE$ and $DC$, b) $I$ is the intesection of $CE$ and $DG$. I will only show a), item b) is similar.

Let $H^\prime$ be intersection of $FE$ and $DC$ (we will show that $H = H^\prime$). Denote $\angle H^\prime FD = \xi$, $\angle JFD = \mu$, $\angle JCE = \nu$. Using the fact that $\angle EFJ = \angle FEJ$, it is easy to establish that $\mu + \nu = 2\xi$ (actually $\mu = \xi + \angle JFE, \nu = \xi - \angle FEJ$). On the other hand one can calculate that $\angle CJF = \mu + \nu$, $\angle CH^\prime F = 2\xi$. Hence $F, H^\prime, J, C$ are on a same circle. Similarly you can show this for $D, H^\prime, J, E$, and the claim follows.

Next, note that $D, H, I, E$ are on a same circle. This means that $\triangle HCI$ is similar to $\triangle ECD$. Therefore: \begin{align*} \angle DAC + \angle ACD &= \angle CDE = \angle HIC = \pi - \angle H K C = \angle KHC + \angle ACD, \end{align*} where the third equality is due to the fact that $K, H, I, C$ are on a same circle. Hence $\angle DAC = \angle BAC = \angle KHC$. But $\angle KHC = \angle KLC$ (because once again $H, K, C, L$ are on a same circle). Thus $\angle KLC = \angle BAC$ and we are done.

  • $\begingroup$ Very clever! I would have never done it. Thanks Sasha! $\endgroup$ – user559615 Aug 3 '18 at 21:49
  • $\begingroup$ How can I accept both the answers? They are both beautiful! $\endgroup$ – user559615 Aug 3 '18 at 21:50

Let us show first the following:

Let $\Delta ABC$ be a triangle. MSE Problem 2871270

We construct $D,E$ on $AB$ so that the triangles $\Delta CAE$, $\Delta CBC$ are isosceles.

The parallel to $CE$ through $D$ intersects $AC$ in $F$.

The parallel to $CD$ through $E$ intersects $AB$ in $G$.

In particular each triangle / trapez in the following list is isosceles: $ACE$, $AFD$, $CEDF$; $BCD$, $BGE$, $CDEG$.

Since an isosceles trapez is inscribed in a circle, the points $C,D,E,F,G$ are on a circle. Let $J$ be its center. Then $J$ is on the perpendicular bisector of all the mentioned $3$-gons and $4$-gons w.r.t. suitable sides, in particular, $J$ it is on the angle bisectors from $A,B$ in $\Delta ABC$. So it is also on the remained angle bisector from $C$. We record: $$JC=JD=JE=JF=JG\ .$$ Let the circles $(CJF)$ and $(CJG)$ intersect the circle $(JDE)$ in the points $H$, respectively $I$.

Then we have the following colinearities:

  • $F,H,E$
  • $C,H,D$
  • $A,H,J$
  • $G,I,D$
  • $C,I,E$
  • $B,I,J$

Proof: The trapez $FDEC$ is isosceles, so the angles in $J$ of the two isosceles triangles $\Delta JDE$, $\Delta JCF$ are equal, being opposite to equal chords. In particular, the four angles (marked in green) at the bases / chords $DE$ and $CF$ are equal.

Let us show that $F,H,E$ are on a line. For this we compute the sum of he angle $\widehat{FHE}$. $$ \begin{aligned} \widehat{FHE} &= \widehat{FHJ}+\widehat{JHE}\\ &= (180^\circ-\widehat{FCJ})+\widehat{JDE}\\ &=180^\circ\ , \end{aligned} $$ because the two "green angles" $\widehat{FCJ}$, $\widehat{JDE}$ cancel each other in measure. We have shown a first colinearity.

Let us show by copy+paste+change that $C,H,D$ are on a line. For this we compute the sum of he angle $\widehat{CHD}$. $$ \begin{aligned} \widehat{CHD} &= \widehat{CHJ}+\widehat{JHD}\\ &= \widehat{CFJ}+(180^\circ-\widehat{JED})\\ &=180^\circ\ . \end{aligned} $$

The triangles $\Delta JHC$ and $\Delta JHE$ are equal (well, congruent, but it is longer), because $JH$ is common side, $JC=JE$, and there are equal red angles in $H$. So $HC=HE$. This means, that $H$ is also on the perpendicular bisector of the side $CE$ in $\Delta ACE$, where $A,J$ live on.

Per symmetry, the remained three colinearities are also proven.


To finish, we consider now also the points $K,L$. In the conditions of the previous result, we continue:

$J$ is the orthocenter of the triangle $CHI$

Let $K,L$ be the points of intersection of the circle $(CHI)$ with the lines $AC$, respectively $BC$. MSE Problem 2871270, second picture

Then we have $$ \begin{aligned} HK &= HF = HD\ ,\\ IL &= IG = IE\ , \end{aligned} $$ and there is the congruence of quadrilaterals $$ DEIH=KLIH\ .$$

Finally, $\Delta CAB\sim\Delta CLK$.

Proof: $HJ$ is a segment on the the perpendicular bisector of the side $CI$ in $\Delta HCI$. So $J$ is on the height from $H$ of this triangle. By analogy, it is also on the height from $I$ of it, so it is the orthocenter in $\Delta CHI$. As a corollary, it is on the third height, so $CJ$ is perpendicular on $HI$. (This will soon help us to compute angles.)

Let us further show $\Delta HKF$ isosceles in $H$ by comparing the angles at the basis $KF$. Let us note in passing that $AFJE$ is inscriptible, because of the two "green angles", interior and exterior to it, in $E$ and $F$. So $\angle HFJ= \angle EFJ=\angle EAJ$ of measure $\frac 12\hat A$, $\hat A$ being the angle in $A$ of the initial triangle $\Delta ABC$. Now we start a longer computation, that joins $K$ and the needed angle to points introduced earlier and earlier: $$ \begin{aligned} \widehat{HKF} &=\widehat{HIC}\qquad\text{(by construction of $K$ on $(CHI)$}\\ &=90^\circ - \widehat{JCI}\\ &=90^\circ - \widehat{JCG}+\widehat{ICG}\\ &=\frac 12(180^\circ - \hat C)+\widehat{ICB}\\ &=\frac 12(\hat A + \hat B)+\widehat{ICB}\\ &=\frac 12\hat A + (\widehat{IBC}+\widehat{ICB})\\ &=\frac 12\hat A + (180^\circ-\widehat{EIB})\\ &=\frac 12\hat A + \widehat{JDE}\\ &=\widehat{HFJ}+\widehat{JFC}\\ &=\widehat{HFC}=\widehat{HFK}\ . \end{aligned} $$ So $\Delta HFK$ is isosceles. We get $HK=HF$ and further $=HD$. The other equality is fulfilled by analogy / by symmetry.

Consider now the quadrilaterals $$ DEIH\qquad\text{and}\qquad LKHI\ .$$ They are inscribed in circles of the same radius, $(CHI)$ and $(JDE)$. (I first saw this in the 5.th class as Problem of the 5L coin. In this second i do not have the reference in English...)

Now we start from $IH$ and construct in the one and the other circle segments $HD=HK$. The triangles $\Delta IHD$ and $\Delta IHK$ are equal. Same for $\Delta HIE$, and $\Delta HIL$ in the other side of the picture. This shows the wanted equality of quadrilaterals.

We are now in position to compute angles in $\Delta CKL$, for instance: $$ \begin{aligned} \widehat{CLK} &=\widehat{CLI}-\widehat{KLI}\\ &=(180^\circ-\widehat{CHI})-\widehat{DEI}\\ &=\widehat{DHI}-\widehat{DEI}\\ &=180^\circ-\widehat{DEI}-\widehat{DEI}\\ &=180^\circ-\widehat{AEC}-\widehat{ACE}\\ &=\hat A\ . \end{aligned} $$ $\square$

Note: It was not my intention to get to the final the shortest path, rather i wanted to "show first everything in the given constellation"... The solution avoids intentionally inversion and metric relations.

  • $\begingroup$ Amazing! I don't have other words ("I don't have other words" had to be written, because the comments box does not allow to write only "Amazing!"). Thanks Dan! $\endgroup$ – user559615 Aug 3 '18 at 21:48
  • $\begingroup$ How can I accept both the answers? They are both beautiful! $\endgroup$ – user559615 Aug 3 '18 at 21:51
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    $\begingroup$ Thanks for posting the beautiful problem, such discovered pearls are always a source of inspiration! I tried to do my best to not formalize the beauty in a solution corner, but rather enrich it. (I do not need the points, it is enough for me to know you followed the path, and enjoyed the story!) $\endgroup$ – dan_fulea Aug 3 '18 at 22:14
  • $\begingroup$ I think it is wonderful, Dan, much more than the problem! I also appreciate the pictures, since I am more "visual" than algebraic. Great answer! (as always, btw). $\endgroup$ – user559615 Aug 3 '18 at 22:20
  • $\begingroup$ About the problem, to me it looks a bit like the golden ratio for the rectangle. The "place" where you perform the cut (i.e. the segment $KL$) makes the cut part similar to the whole triangle, but not in a trivial way (e.g. like to cut the triangle along a parallel of one side)... $\endgroup$ – user559615 Aug 3 '18 at 22:26

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