Questions tagged [triangles]
For questions about properties and applications of triangles.
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How to find a point within a triangle from perpendiculars taken from the edges
Given a point on each edge of the triangle, where, if you were to take the perpendicular of the edge from that point, all 3 perpendiculars would intersect a single point, how could I find the position ...
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transforming a polynomial function
I was exploring transforming polynomials (sorry if this is the wrong term). Essentially, I found a way to rewrite polynomials in different equivalent forms analogous to changing a quadratic from ...
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Area of the wet part of a horizontal cylinder
So, my math teacher gave me an interesting problem on mensuration.
Given, a cylinder of Height $H$ and radius $r$ is filled with water upto height h. Then the cylinder is pushed and it lies down ...
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In expectation, what is the area of a triangle formed by three points chosen randomly on the surface of the earth?
Three points are randomly chosen on the surface of a sphere. They are connected to each other by great arcs to form a (curved) triangle. What is the expected value of the area of the triangle?
We know ...
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Show that given three points are colinear with the circumcircle.
This is a practice math olympic problem from Peru. It is originally in Spanish as following:
I tried to translate as following:
$\triangle ABC$ and $D$ is on $BC$. Let $AD$ intersect the circumcircle ...
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Area of the triangle inside the triangle
Area of each shape in the triangle is written. What is the area of the
shaded region?
Based on my search, $\dfrac{S_{\triangle MNP}}{S_{\triangle ABC}}$ can be calculated by Routh's Theorem. assuming ...
- 6,794
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Finding the Relationship Between $∠GCF$ and $α$ in a Rhombus with an Isosceles Triangle
I am working on a problem involving a rhombus $ABCD$, where point $E$ lies on side $BC$. Triangle $AEF$ is an isosceles triangle with $AE = EF$, and $∠AEF = ∠ABC = α$, where $α$ is at least $90°$. ...
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Two right triangles inside a bigger triangle
In the given figure, ABC is a triangle.
We know that:
AF $\perp$ DC
DE $\perp$ BC
AD = 2 * DB
DC = 8 cm
BC = 12 cm
What is $\dfrac{\text{AF}}{\text{DE}}$?
I started by labeling the sides:
Then ...
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Would every triangle have 3 concurrent cevians such that they intersect each other at $60^{\circ}$?
In an equilateral triangle, the three angle bisectors meet at the incenter. They intersect with each other forming angles of $60^{\circ}$ with each other. Will every triangle have 3 concurrent ...
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One angle in a triangle is twice the other. How to generalize the fact about its sides?
One angle in a triangle is twice the other. If I drop a perpendicular from the vertex of the third angle of the same triangle, then I suspect that the projection of the sides opposite to these angles ...
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Implication about the hypotenuse of a triangle being multiple of $\sqrt{2}$
If the hypotenuse of a triangle is a multiple of the square root of two, does it necessarily imply that the two legs of the triangle are equal? What about if both sides are integers?
If $a^2+b^2 = 2k^...
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Proving triangle midsegment theorem without quadrilaterals or similarity
I'm here with a question on a basic fact presented to you below:
The midline of a triangle is parallel to the third side of that triangle and half its length.
There is a known proof listed here on ...
- 848
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Counting number of Triangles with Integer-coordinate Vertices in the xy-plane
How many triangles with positive area are there whose vertices are points in the $xy$-plane whose
coordinates are integers $(x, y)$ satisfying $1 \le x \le 4$ and $1 \le y \le 4$?
There are $16$ ...
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Area of tight-angled $\triangle POB$ given extensions of $OP,BP$ to circle centred at $O$ through $B$?
We have a triangle $(\triangle POB)$ within a semicircle. $OP$ and $BP$ are extended to $OA$ and $BQ$. $AP = 5$ and $PQ = 7$. What is the area of the triangle?
It's a problem I stumbled upon on ...
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Find length of 3rd leg of a triangle (that is not a right triangle)
Assume I have any triangle $\triangle ABC$. I know that given the lengths two sides of the triangle and angle between them, I can find the length of the third side. In other words, given values of $AB$...
