Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

ABCD is a cyclic quadrilateral and the points A , B and C form an equilateral triangle .

Then what is the sum of the length of line segments DA and DC? What property should i use to get the value of DA and DC.

I need a hint to start the solution.

Thanks in advance.

share|improve this question
    
What does PQRS have to do with anything? Where is D in relation to ABC? –  anon Apr 1 '12 at 8:44
    
@robjohn i am sorry.edited the question. –  vikiiii Apr 1 '12 at 8:46
    
Hint: Ptolemy's theorem. –  dtldarek Apr 1 '12 at 8:56

3 Answers 3

Let's denote :

$|AB|=|BC|=|AC|=a$

$|CD|=b$

$|AD|=c$

Since quadrilateral is cyclic and $\Delta ABC$ is equilateral it follows that :

$\frac{1}{2}(a^2+bc)\cdot \sin 60^{\circ} =\frac{a^2 \sqrt 3}{4}+\sqrt{s(s-a)(s-b)(s-c)}$

where $s=\frac{a+b+c}{2}$

share|improve this answer

The given information does not suffice to compute $|DA|+|DC|$, as $D$ can be any point on the shorter arc between $A$ and $C$. All you can say is the following: In the triangle $ADC$ the angle at $D$ is $120^\circ$, whence by the cosine theorem on has $$|DA|^2+|DC|^2 + |DA|\ |DC|=|AC|^2=3 r^2\ ,$$ where $r$ is the radius of the circumscribing circle.

share|improve this answer

Use Ptolemy's theorem,

Let AB = BC = AC = a Then by Ptolemy's theorem, AC x BD = AB x CD + BC x AD a x BD = a ( DA + DC ) BD = DA + DC

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.