Monthly Archives: April 2013

Three icecream cones and a line

Infinitely Curious friends,

Every once in a while I meet a mathematical theorem that puts a smile on my face because of its simplicity and “cuteness”. In today’s VERITAS I will tell you about one such mathematical theorem. I read it about a week back and still continues to make me feel happy. So I decided to share this theorem with you.


Here is what it is: Take a sheet of paper and make three circles of different sizes. The circles should be separate from each other. Let’s call these three circles A, B and C. Now draw two tangents to A and B. In other words draw one line that touches A and B but does not go inside it – this is a tangent to A and B. Now draw another tangent to A and B. So we have two common tangents to A and B. These tangents will cross( intersect) somewhere. Call this point P1. At this time it will look like an ice cream cone holding two scoops of ice cream of different sizes. Now draw two tangents to A and C. These tangents will cross at a point- call it P2. Similarly draw two tangents to B and C- these will intersect at P3. The cute mathematical fact is that P1, P2 and P3 lie on a straight line. Isn’t that so simple and yet so profound. See the attached image for a visual representation. It will look like three ice cream cones holding scoops A, B and C in pairs. The tips of the cones lie on a straight line.

This theorem is called Monge’s theorem. This theorem is valid even if the circles intersect.

Now a little bit about the mathematician who discovered this interesting mathematical theorem. Gaspard Monge was a french mathematician. Not many of us know his name but all the engineers of this world have used his invention- descriptive geometry. The engineers among you would remember the first year classes on technical drawing( I remember them with horror ). In technical drawing we learnt to represent three dimensional objects on 2D paper using different views and projections- eg top view, front view etc. These techniques were invented by Monge and are known as “descriptive geometry” in mathematics. After the French revolution Monge became very influential under Napoleon’s rule. He was instrumental in the setup of the Ecole Polytechnique- which is one of the great institutions of mathematics and sciences. He also was a part of Napoleon’s expedition to Egypt. In an earlier VERITAS( written in the year 2000), I had written in detail about Napoleon’s interest in mathematics and his trip to Egypt in which he had taken a number of scientists and mathematicians. In that VERITAS I had also written that Napoleon himself was a mathematician and even has a theorem by the name of Napoleon’s theorem. For details read VERITAS: Napoleon’s academic conquest of Egypt, 25th June 2000 (

Friends, I feel that there are three attitudes among most people about mathematics. The first two are “bad” attitudes: some people fear mathematics. Some others are arrogant about their mathematical abilities. Both these attitudes are caused by our educational system which uses mathematics to judge the intelligence of people. Mathematics becomes a scary competition in which 95% people are declared “bad at math” and 5% are considered “good in math”. This causes the 95% of people to develop a fear and 5% to develop an arrogance of their mathematical abilities. Both these attitudes are a result of comparison and comparisons can never result in true knowledge. There is a third attitude which is the correct one. It is about amazement and love for the beauty of mathematics. This attitude does not compare. It is about enjoying the infinite relationships between numbers and shapes. This attitude promotes a lifelong interest in mathematics and an appreciation of its beauty. This is the attitude that we need to promote in our kids. For our kids mathematics should be a way of looking at this amazing world and not about how much his friend scored.

Before I end this VERITAS here is a little quote by G.H Hardy( the person who first recognized Ramanujan’s genius):

The mathematician’s patterns, like the painter’s or the poet’s must be
beautiful; the ideas, like the colors or the words must fit together in a
harmonious way. Beauty is the first test: there is no permanent place in
this world for ugly mathematics.


Go wondrous creature, mount where science guides
go measure earth, weigh air, state the tides,
instruct the planets in what orbs to run
correct old time, regulate the sun