# Tagged Questions

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### To construct a right triangle given the hypotenuse and sum of two legs

NOTE: I want a hint only. A compass and a straightedge construction:Given a hypotenuse and the sum of lengths of the legs,we need to construct a right triangle. MY TRY: From any ray $BE$, ,let ...
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### Draw a picture of a cube with edges a+b, and show it cut by planes that divide each edge into a segment of length a and a segment of length b.

I am reading through 4 pillars of geometry and I need some help with this question. Draw a picture of a cube with edges a+b, and show it cut by planes (parallel to its faces) that divide each edge ...
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### Maximize the inradius given the base and the area of the triangle

BdMO 2013 Secondary: A triangle has base of length 8 and area 12. What is the radius of the largest circle that can be inscribed in this triangle? Let $A,r,s$ denote the area,inradius and ...
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### Perpendicular lines inside and outside a circle

No trigonometry allowed. Let $\Delta ABC$ be inscribed inside a circle.Let $P$ be a point on the circle.Let $PD$ and $PE$ be perpendiculars on on $BC$ and $AC$ respectively.Let $DE$ when extended ...
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### Heptagonal tesselations

Are there any tesselations of the Euclidean plane that use only regular polygons such that one of them is a heptagon? If so, what is the tesselation that uses the fewest different types of polygon ...
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### which axiom(s) are behind the Pythagorean Theorem

There are many elementary proofs for the Pythagorean Theorem, but no matter they use areas, similarities, even algebraic proofs, it is not straightforward to tell why it is true tracing back to the ...
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### Prove $\sin^2 A = \sin^2 B \sin^2 C - 2\sin B \sin C \cos A$

I am asking for help with this proof: Given $\triangle ABC$. Prove that $\sin^2 A = \sin^2 B+ \sin^2 C - 2\sin B \sin C \cos A$
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### Area of a equilateral triangle given distances of a point in the triangle from the vertices [closed]

A point $D$ inside an equilateral triangle $PQR$. $D$ is located at a distance of $3$cm, $4$cm and $5$cm respectively from $P$, $Q$ and $R$. What is the area of the triangle $PQR$?
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### Given a triangle find the length of BC

Given an ABC triangle , if $AB+AD=4$, find the length of BC.
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### Moscow Math Olympiad 1973

In every polyhedron there is at least one pair of faces with the same number of sides. Solution: Let $N$ be the greatest number of sides in a face of a given polyhedron. Then the number of ...
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### Area of intersection between square and annulus

The annulus' larger radius is $1$, smaller radius is $r>0$, and center is $(0,0)$. The square's sides are parallel with the axes, the lower left corner's coordinates are $(a,b)$, and the upper ...
NOTE: I am NOT looking for a full answer,just a hint. Last problem on this question. BdMO 2013 Chittagong: Let $ABCD$ be a rhombus.Let $G$ be a point outside the rhombus such that GE is ...