All Questions
Tagged with triangles euclidean-geometry
1,383
questions
2
votes
4
answers
107
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Is there is a formula to calculate the coordinates of the orthocenter of a triangle?
I'm trying to find the coordinates of the orthocenter (the intersection point of all altitudes) of a triangle given its vertices' coordinates $A=(x_1, y_1), \ B=(x_2, y_2) , \ C=(x_3, y_3)$.
I ...
- 6,565
2
votes
1
answer
73
views
Determine the angle $\angle DEC$ in a triangle (Euclidean Geometry)
Any ideas how to find the angle $\angle DEC$ in the following situation shown in the image:
In the above figure we have that $\angle BAC = 90, \angle ABD = \alpha, \angle DBC = 2\alpha$, and $\angle ...
- 2,964
1
vote
2
answers
41
views
Alternate Proof for Sum of Sides of a Triangle Inequality
I recently stumbled upon an idea for a proof for the sum of two sides of a triangle inequality.
Note that I am just a high school student and feel free to correct me wherever if I am wrong.
Statement/...
- 13
1
vote
0
answers
37
views
Proof of Thomson cubic pivotal property without coordinates
The Thomson cubic is defined as the cubic going through A,B,C, the three side midpoints, the three excenters. Is there a way to prove its pivotal property (any two isogonal conjugates on it have a ...
- 264
3
votes
2
answers
292
views
What is the minimum value of $a+b-c$ in a triangle with a fixed area?
Let $\Delta$ be the fixed area of a triangle inscribed inside on a fixed circle of radius $R$. The sides of the triangle $(a,b,c)$ are unknown. We want to estimate a the lower bound of the triangle ...
- 20.8k
0
votes
1
answer
77
views
Find the segment BT in the triangle inscribed below
In the figure, $AB.BC = 60$ and $BT.TP = 40$.
Calculate BT with B and T tangency points.
(Answer:$2\sqrt5$)
I try:
$AT.TC = BT.TP \implies AT.TC = 40$
$AM.AB = AT.AC$
$AT^2 = AM.AB \implies AT^2 = AT....
- 7,031
2
votes
1
answer
79
views
Isosceles triangle calculations
We have that if $𝑆 ⊂ ℝ^2$ is a set of $𝑛$ points in the plane, with no three points on a common line, then there exists a point $𝑎 ∈ 𝑆$ that determines at least $(𝑛 − 1)/3$ distinct distances to ...
- 282
0
votes
2
answers
110
views
Find the missing angle in the triangle below if the sum of two sides is given
I'm having a hard time solving this problem, so if anyone would be kind enough to point me in the right direction, I'd be forever grateful!
Find the value of the the missing angle in the triangle ...
- 41
4
votes
3
answers
186
views
Find the sum $PC^2+PD^2$ in the trapezoid inscribed below
If there is a semicircle of diameter $AB$ in which an isosceles trapezoid $ABCD$, ($AB \parallel CD$) is inscribed. On $AB$, we take a point "$P$" such that $PA^2 + PB^2 = 5^2$.
Calculate: $...
- 7,031
4
votes
3
answers
147
views
Can the center of circumscribed circle in a triangle be on the incircle?
Besides the obvious answer of an isosceles right triangle, can there be other triangles where the center of its circumscribed circle is located on the perimeter of its incircle?
I tried using the ...
1
vote
2
answers
84
views
A geometry problem involving three altitudes of a triangle.
I think this is a well-known result in plane geometry but I don't remember how to solve it. So I decided to post it here, hoping that I would get a hint or a solution if possible.
Let $ABC$ be an ...
- 499
1
vote
1
answer
106
views
Tangent problem involving orthocenter and circumcenter in a triangle
Let $ABC$ be an acute triangle inscribed the circle $(O)$ with three altitudes $AD, BE, CF$, orthocenter $H$. The tangent of $(O)$ at $A$ intersects $BC$ at $S$. Let $M$ be the midpoint of $BC$, $K$ ...
- 499
3
votes
4
answers
174
views
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
1
vote
4
answers
168
views
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 $90°$. ...
- 381
2
votes
1
answer
66
views
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 ...
- 848