The Problem Let ABC be a triangle in which AB = AC and let I be its in-centre. Suppose BC = AB + AI. Find ∠BAC. Big Ideas For any triangle ABC, \( \frac{\sin A}{a} = \frac{\sin B } {b} = \frac {\sin C }{c} \). Addendo: If \( \frac{a}{b} = \frac{c}{d} \) then each of […]
Recently, French mathematician Cedric Villani's team came up with '21 measures for the teaching of Mathematics'. I read through the report, with great curiosity. I happily noted that Cheenta's Thousand Flowers program has already implemented some of his recommendations.
Pythagoras theorem can be extended! What happens if the triangle is obtuse-angled (instead of right-angled?) We explore the idea by using a problem from Math Olympiad.
Problem Let ABC be an acute-angled triangle, let D, F be the mid-points of BC, AB respectively. Let the perpendicular from F to AC and the perpendicular at B to BC meet in N. Prove that ND is equal to circum-radius of ABC. Theorems and tools The discussion uses the following Theorems: Midpoint Theorem: The line […]
TIFR 2017 Math Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India's premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects. The image is a front cover […]
This is a problem from ISI B.Stat-B.Math Entrance Exam 2018, Subjective Problem 7. It is based on Bases, Exponents and Role reversals. I.S.I. Entrance 2018 Problem 7 Let $(a, b, c)$ are natural numbers such that $(a^{2}+b^{2}=c^{2})$ and $(c-b=1)$. Prove that(i) a is odd.(ii) b is divisible by 4(iii) $( a^{b}+b^{a} )$ is divisible by […]
Problem In a quadrilateral $ABCD$. It is given that $AB=AD=13$, $BC=CD=20$, $BD=24$. If $r$ is the radius of the circle inscribable in the quadrilateral, then what is the integer close to $r$? Hint 1: First, notice that the quadrilateral is a kite. Diagonals of a kite bisect each other (Prove this!) If $X$ is the point […]