### Some Wandering Through Orthic Triangles

Hi, I am Emon and I it's been long since I posted last. Today I will try to give you some ideas on how to work with a special type of triangle, known as "Orthic Triangles". In this post, I will mainly focus on problem-solving, but still, let me first give you some ideas on what exactly it is and what properties does it have...

Definition (Orthic Triangle). Let $ABC$ be a triangle and let $D, E, F$ be the foot of the perpendiculars from $A, B, C$ to $BC, CA$ and $AB$, respectively. Then, $\triangle DEF$ is known as the orthic triangle of $\triangle ABC$.

Lemma $1$ (Orthic Triangle). If $\triangle DEF$ is the orthic triangle of $\triangle ABC$ with orthocenter $H$, then the following conditions are satisfied :

$(i)$ $AEHF$ is a cyclic quadrilateral with circumdiameter $AH$.
$(ii)$ $BCEF$ is a cyclic quadrilateral with circumdiameter $BC$.
$(iii)$ $H$ is the incenter of $\triangle DEF$.

Lemma $2$. $\angle ABE = \angle ADE$ and $\angle ACF=\angle ADF$.
(We can prove this with the help of some inscribed angles)

Okay, so as I had mentioned earlier, we shall now move into some problem-solving without further delay.

Problem $1$. In the acute triangle $ABC$, $A',B',C'$ are the feet of the perpendicular from vertices $A,B,C$ onto the lines $BC,CA,AB$ respectively. Find all possibilities of the angles of the triangle $ABC$ for which the triangles $ABC$ and $A'B'C'$ are similar.

Solution.

We first have that $ABA'B'$ is cyclic. Hence, $\angle B'A'C=\angle BAC$. Similarly, since $ACA'C'$ is cyclic, we have, $\angle C'A'B=\angle BAC$.
Thus, we have, $\angle BAC=\angle B'A'C'=180^\circ-2\angle BAC$, since $\triangle ABC\sim \triangle A'B'C'$, which implies that $\angle BAC=60^\circ$.
Similarly, we get, $\angle ABC=\angle BCA=60^\circ$, so that $\triangle ABC$ is equilateral.
Hence, we can conclude that all equilateral triangles $ABC$ satisfy the given conditions.

Problem $2$ (IberoAmerican SL $1995/1$). In an acute triangle $ABC$, the distance from the orthocenter to the centroid is half the radius of the circumscribed circle. $D, E$, and $F$ are the feet of the altitudes of triangle $ABC$. If $r_1$ and $r_2$ are, respectively, the radii of the inscribed and circumscribed circles of the triangle $DEF$, determine $\frac{r_1}{r_2}$.

Solution. Let $H$ be the orthocentre, $O$ the circumcentre, $N$ be the nine point centre and $G$ be the centroid of $\triangle ABC$.
Now it is given that $HG= \frac{R}{2}=2GO$. So $HO =\frac{3R}{4}$. Hence, $NH= \frac{3R}{8}$ because $N$ is the midpoint of $OH$.
Now, since $(DEF)$ is the nine point circle of $\triangle ABC$, $r_2 =\frac{R}{2}$ (the nine point radius is half the circumradius by homothety.)
But, $H$ is the incentre of $\triangle DEF$ by Lemma $2(iii)$. So, by Euler's formula $NH^2= (r_2)^2 -2r_1r_2$, as $N$ is the circumcentre of $DEF$ and $H$ is the incentre. Now putting the value of $NH$ and $r_2$, we get that $\displaystyle{r_1 =\frac{7R}{64}}$. Hence the ratio is $\boxed{\frac{7}{32}}$.

Problem $3$. Show that the perimeter of the orthic triangle of a triangle is less than twice of any altitude.

Solution. Let $D, E, F$ be the foot of the altitudes on $BC, CA, AB$ from $A, B, C$, respectively.
We first observe that the perimeter of the orthic triangle $DEF$ of $\triangle ABC$ is $4R\sin A\sin B\sin C$, where $R$ denotes the circumradius of $\triangle ABC$.
Now, $2AD=2(2R\sin B\sin C)=4R\sin B\sin C$.
But, since $\sin A < 1$, we have, $DE+EF+FD=4R\sin A\sin B\sin C<4R\sin B\sin C=2AD$. Similarly, we have, $DE+EF+FD<2BE$ and $DE+EF+FD<2CF$, and hence we may conclude that, the perimeter of the orthic triangle $DEF$ is less than twice of any altitude $AD, BE, CF$, as desired.

That's all from my part, I guess. I would leave a few problems for you to try...

Problem $4$ (Ukraine National MO $2000\ 11.3$). Let $AA_1, BB_1, CC_1$ be the altitudes of the acute triangle $ABC$. Denote by $A_2, B_2, C_2$ the points of tangency of the circle inscribed in triangle $A_1B_1C_1$, with sides $B_1C_1$, $C_1A_1$, $A_1B_1$ respectively. Prove that the lines $AA_2$, $BB_2$, $CC_2$ intersect at one point.

Problem $5$. Let $\triangle XYZ$ be the orthic triangle of the orthic triangle of $\triangle ABC$. Show that $AX,BY,CZ$ meet on the Euler line of $\triangle ABC$.

Problem $6$. Let $\triangle{H_A H_B H_C}$ be the orthic triangle of $\triangle{ABC}$. Prove that the line perpendicular to $\overline{H_A H_B}$ intersecting $C$, the line perpendicular to $\overline{H_B H_C}$ intersecting $A$, and the line perpendicular to $\overline{H_C H_A}$ intersecting $B$ concur at the circumcenter of $\triangle{ABC}$.

Problem $7$. Let $\triangle{M_A M_B M_C}$ and $\triangle{H_A H_B H_C}$ be the medial and orthic triangles, respectively, of acute scalene triangle $\triangle{ABC}$ with shortest side $\overline{AB}$. The medial and orthic triangles intersect at four points, two of which lie on $\overline{H_A H_B}$. Prove that $M_A$, $M_B$, and those two points lie on a circle.

Problem $8$. Let $\triangle ABC$ be an acute angled triangle with $AB<AC$. Let $D,E,F$ be the intersections of angle bisector of $\angle A$, $\angle B$ and $\angle C$ with opposite sides of $\triangle ABC$. Let $\triangle H_AH_BH_C$ be the orthic triangle WRT $\triangle ABC$. Let $M$ be the midpoint of $BC$. Let $H_BH_C \cap EF$ $=$ $P$. Assume, $P$ to be inside $\triangle ABC$. Prove, $MP$ passes through one of the intersections of $\odot (H_AH_BH_C)$ and $\odot (DEF)$.

Okay, so that's it. There's nothing in the theory of orthic triangles as such. It can be better understood by solving more and more problems. I have tried to give you some ideas on how to solve questions related to it, and you should try some more on your own to get a good grasp on it. Hope you have enjoyed.
Bye bye until next time...

Emon.

### Constructions in Number Theory

Hi, I am Emon, a ninth grader, an olympiad aspirant, hailing from Kolkata. I love to do olympiad maths and do some competitive programming in my leisure hours, though I take it seriously. I had written INOI this year. Today, I would be giving you a few ideas on how to Construct in Number Theory . Well, so we shall start from the basics and shall try to dig deeper into it with the help of some important and well-known theorems and perhaps, some fancy ones as well. Okay, so without further delay, let's start off... What do we mean by "Constructions"? If noticed, you can see that you often face with some problems in olympiad saying, "... Prove that there exists infinitely many numbers satisfying the given conditions" or "... Prove that there exists a number satisfying the above conditions." These are usually the construction-problems .  For example, let's consider a trivial example : Problem. Prove that there exist infinitely many integers $a$ such th

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