Try this beautiful Problem on Geometry based on Interior Point of a Triangle from PRMO-2017
Let $P$ be an interior point of a triangle $A B C$ whose side lengths are $26,65,78 .$ The line through P parallel to BC meets AB in K and AC in L. The line through P parallel to CA meets BC in M and BA in N. The line through P parallel to AB meets CA in S and CB in T. If KL, MN, ST are of equal lengths, find this common length.
,
Geometry
Triangle
modulus
Pre College Mathematics
Prmo-2017, Problem-24
no solution is possible.
We have to find out the common length .
Clearly PKBT, PMCL and PSAN are parallelograms. Let $P T=K B=x, P M=L C=y$,
$\mathrm{PK}=\mathrm{BT}=\mathrm{z}$ and $\mathrm{KL}=\mathrm{MN}=\mathrm{ST}=\ell$
because $\Delta P T M \sim \Delta A B C$ [as $PT ||AB$,$PM||AC$, therefore interior angles are same, so $\triangle PTM \sim \triangle ABC$]
Since $\Delta \mathrm{PTM} \sim \Delta \mathrm{ABC} $
$\frac{y}{65}=\frac{26-\ell}{26}$
Again, $\Delta \mathrm{NKP} \sim \Delta \mathrm{ABC}$ [ as $NP||AC$, $KP||BC$, therefore interior angles are same, so $\Delta \mathrm{NKP} \sim \Delta \mathrm{ABC}$ ]
$\Rightarrow \frac{\ell-y}{65}=\frac{78-\ell}{78}$
Adding (1)$\&(2)$
$\frac{\ell}{65}=\frac{26-\ell}{26}+\frac{78-\ell}{78}$
$=\frac{78-3(+78-\ell}{78}$
$\Rightarrow 6 \ell=5(156-4 \ell)$
$\Rightarrow 26 \ell=5 \times 156$
$\Rightarrow \ell=\frac{5 \times 156}{26}=30$
But $\ell$ must be less than $26,$ hence no solution is possible.
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]
In 2025, 8 students from Cheenta Academy cracked the prestigious Regional Math Olympiad. In this post, we will share some of their success stories and learning strategies. The Regional Mathematics Olympiad (RMO) and the Indian National Mathematics Olympiad (INMO) are two most important mathematics contests in India.These two contests are for the students who are […]
Cheenta Academy proudly celebrates the success of 27 current and former students who qualified for the Indian Olympiad Qualifier in Mathematics (IOQM) 2025, advancing to the next stage — RMO. This accomplishment highlights their perseverance and Cheenta’s ongoing mission to nurture mathematical excellence and research-oriented learning.