Geometric Inequality

I discovered a nice inequality while playing on Geogebra again. I found that it can be proved quite quickly by making a simple construction.
The triangle ABC, with sides of length a, b and c opposite vertices A, B and C respectively, satisfies a√2 ≥ b+ c. M is the midpoint of AB and P is on BC such that ∠MPA = ∠BCA. Then BC ≥ 2AP.
geo ineq 1.png
Proof
The condition that a√2 ≥ b+ c is merely necessary to guarantee the existence of at least one such P. A derivation of this inequality will be included at the end of this post.
Let Q be the point on AP extended such that AP = PQ.
Triangle ABQ is an enlargement of triangle AMP, since AB = 2AM and AQ = 2AP, so MP is parallel to BQ and ∠MPA = ∠BQA.
So BQCA is a cyclic quadilateral by the converse of the same segment theorem.
geo ineq 2.png
Applying intersecting chords theorem, AP · PQ = BP · PC; that is, AP² = BP · PC
Let BP = x and let PC = y.

But from the AM-GM inequality,

*This is also apparent from rearranging

So

And as BP + PC = BC, this means BC ≥ 2AP as desired.
Equality occurs when

which is when P is the midpoint of BC. In this case, it is easy to show that triangle ABC is right angled at A.

Derivation of condition for existence of at least one such P
Let N be the midpoint of AC. Then triangle ABC is an enlargement of triangle AMN, as AB = 2AM and AC = 2AN, so ∠MNA = ∠BCA.
geo ineq 3
The set of points P such that ∠MPA = ∠BCA = ∠MNA is the set of points located on the circumcircle of triangle AMN, on the same side of AM as N; this is due to the same segment theorem.
Hence, for there to be such a P located on BC, the circumcircle of AMN must touch BC at least once.
Clearly this does not always happen:
geo ineq 5.png
Sometimes there are two such P:
geo ineq 4.png
And sometimes, in the case we will now examine, there is exactly one such P.
geo ineq 6.png
In this case we can apply secant-tangent theorem:




Also





Fix the side BC and consider for which points A there exists exactly one P. a√2 is constant so the locus of A is an ellipse, since the sum of the distances to each of two foci is constant for every point on the locus. Here, the foci are B and C, and the constant sum is a√2.
geo ineq 7.png
For all points A on the ellipse, the circumcircle of triangle AMN is tangent to BC.
We can now consider the cases where the circle AMN is not tangent to BC.
If we move A within the ellipse, the sum of the distances from A to the foci decreases to become less than a√2, and the circumcircle of triangle AMN moves towards BC so that it cuts it twice. Hence there are two such P when a√2 > b+ c.
geo ineq 8.png
If we move A out of the ellipse, the sum of the distances from A to the foci increases to become more than a√2, and the circumcircle of triangle AMN moves away from BC so that it no longer touches it. Hence there is no such P when a√2 < b+ c.
geo ineq 9.png
Therefore, for there to exist at least one such P, we must have a√2 ≥ b+ c.
I am grateful to my friend Jeremy Ho for helping to check this result.

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