Problem 22
Let O be any point inside a triangle ABC. Then, show that:
AO + BO < AC + BC
Here’s a simulation that demonstrates the problem.
Try dragging the points A, B, C, and O. Is AO + BO always less than AC + BC? How can we prove the same?
Solution
I’ll give you two [entirely different] directions to solve this problem.
Triangle Inequality
The sum of two sides of a triangle is always greater than the third side. That is, in a triangle ABC, we have:
AB + BC > CA
BC + CA > AB
CA + AB > BC
To proceed, you’ll have to figure out the triangles where to apply this inequality. You might need to construct a triangle as well. Also, you’ll need to apply this inequality twice [i.e., in two different triangles] and add them to get the required result.
Ellipse
The sum of the distances of any point on an ellipse from the two foci is always constant. This distance is equal to the major axis of the ellipse.
Here’s a simulation that demonstrates this property.
Try dragging the point P on the ellipse. Is PS + PS‘ always fixed [and equal to AB]? You can also drag S and S’ to vary the ellipse.
Now, we have to prove that AO + BO < AC + BC. Can you bring in the above property of the ellipse here? Which points can we take as the foci?
Were you able to solve it using the above hints? Or, did you solve using any other method? Or, do you need some more help? You can reach out at the Telegram group.
Futher Exploration
Can these two sums be equal? If yes, when would that happen?
Problem Source: CutTheKnot
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