3 points split a segment into 4 equal parts wanted

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3 points split a segment into 4 equal parts wanted

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Source: Czech-Polish-Slovak Match Junior 2019, individual p2 CPSJ

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#1 h Jan 20, 2020, 5:00 PM • 2 Y

Y by Adventure10, Mango247

Let $ABC$ be a triangle with centroid $T$ . Denote by $M$ the midpoint of $BC$ . Let $D$ be a point on the ray opposite to the ray $BA$ such that $AB = BD$ . Similarly, let $E$ be a point on the ray opposite to the ray $CA$ such that $AC = CE$ . The segments $T D$ and $T E$ intersect the side $BC$ in $P$ and $Q$ , respectively. Show that the points $P, Q$ and $M$ split the segment $BC$ into four parts of equal length.

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#2 h Feb 11, 2020, 2:45 PM • 1 Y

Y by Adventure10

Let $B'$ and $C'$ be points on $BC$ such that $TB' \parallel AB$ and $TC' \parallel AC$ .
We see that there is a homothety taking $\triangle B'TC'$ to $\triangle BAC$ with scale factor $3$ .
$B'M = \frac{1}{3} BM$ , so $BB' = \frac{2}{3} BM$ . We also notice that $\triangle BDP \sim \triangle B'TP$ .
With $B'T =\frac{1}{3} AB = \frac{1}{3} BC$ , then $BP = 3 PB'$ , so $BP = \frac{3}{4} BB' = \frac{3}{4} \cdot \frac{2}{3} BM = \frac{1}{2} BM$ . Similarly, it can be said that $CQ = \frac{1}{2} CM$ , and the conclusion follows.

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#3 h Dec 26, 2020, 7:25 AM

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We can "coordinate bash" using distance down and distance away from the horizontal projection onto AB. We have (1/3, 1/6), and (1, 0) as TD. They make the line y=-1/4(x-1). It goes through (1/2, 1/8) and so P splits BC into a quarter. similarly Q splits BC into another quarter and M splits it into halves so they are at 1/4, 1/2 and 3/4 and we are done

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