Track / Stage 6 / 221 of 400 #1221 of 1964

Problem 1221

National Olympiad, first round

Geometry Difficulty 6.2 Prove it

A line $l$ is drawn through the centroid of triangle $A B C$ , intersecting sides $A B$ and $B C$ . We will prove that the sum of the distances from points $A$ and $C$ to the line $l$ is equal to the distance from point $B$ to this line.

Given (Fig. 37): $\triangle A B C, O$ - centroid of $\triangle A B C, l \cap[A B]$ , $l \cap[B C],\left(A A_{1}\right) \perp l,\left(B B_{1}\right) \perp l,\left(C C_{1}\right) \perp l,\left\{A_{1}, B_{1}, C_{1}\right\} \in l$ .

To Prove: $\left|A A_{1}\right|+\left|C C_{1}\right|=\left|B B_{1}\right|$ .

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Proof. Let's consider a purely geometric proof.

Draw the median $BD$ of triangle $ABC$ . Since point $O$ is the centroid of $\triangle ABC$ , then $O \in [BD]$ . Drop a perpendicular $DD_1$ from point $D$ to line $l$ . Since $(AA_1) \perp l$ and $(CC_1) \perp l$ , then $(AA_1) \parallel (CC_1)$ , i.e., quadrilateral $AA_1CC_1$ is either a rectangle (if $l \parallel (AC)$ ) or a trapezoid (if $l \perp (AC)$ ).

If $l \parallel (AC)$ , then from the similarity of triangles $OBB_1$ and $ODD_1$ it follows that $|BB_1| = 2|DD_1|$ . Since in the considered case $|DD_1| = |AA_1| = |CC_1|$ , then $|AA_1| + |CC_1| = |BB_1|$ .

If $l \perp (AC)$ , then since $|AD| = |DC|$ and $(DD_1) \parallel (AA_1)$ , then $[DD_1]$ is the midline of trapezoid $AA_1C_1C$ , i.e.,

$|DD_1| = \frac{|AA_1| + |CC_1|}{2}$

Since $(DD_1) \perp l$ , $(BB_1) \perp l$ , and $\widehat{BOB_1} = \widehat{DOD_1}$ , then $\triangle BOB_1 \sim \triangle DOD_1$ and therefore

$\frac{|DD_1|}{|BB_1|} = \frac{|OD|}{|OB|} = \frac{1}{2}$

hence,

$|DD_1| = \frac{1}{2}|BB_1|$

Comparing the expressions for $|DD_1|$ from equations (33.1) and (33.2), we get:

$|AA_1| + |CC_1| = |BB_1|$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.