Thursday, May 6, 2010

Solution: Cyclic quadrilateral

Original Problem:

First Solution

Let $AB=AD=x, BC=y$ and $CD=x+z$ with $y < z$.

Let $\theta = \angle ADC$ so $\angle ABC = 180^o-\theta$. It's easy to see that because $CD > BC$ then $\angle ABC > \angle ADC$ thus $0 < \theta < 90^o$

Now $AC^2 = x^2 + y^2 + 2xy \cos \theta = x^2 + (x+z)^2 - 2x(x+z)\cos \theta$, simplify it to get:
$\cos \theta = \frac{(x+z)^2-y^2}{2x(x+y+z)} = \frac{x+z-y}{x}$

But since $\theta$ is an acute angle, $\theta < 60^o \iff \cos \theta > 1/2$
So we need to show
$x+z-y > x$ which is true because $z > y$

Second Solution

Let $R$ be a point on $CD$ such that $CB = CR$ (see the picture).

Since $AB=AD$, then $CA$ is an internal angle bisector, and thus $\triangle ABC$ are congruent to $\triangle ARC$. That means $RD = CD - CR = CD - CB > AB = AD$.
Also, $AD = AB = AR$.
So $\triangle ARD$ is an isosceles where $AD = AR$ and $RD > AD$, which means that $\angle ADR < 60^o$ and thus $\angle ABC > 120^o$

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