In a triangle ABC, let D and E be the intersections of the bisectors of the angles ABC and ACB with the sides AC and AB, respectively. Knowing that the measures in degrees of the angles BDE and CED are equal to 24 and 18, respectively, calculate the difference in degrees between the measures of...
1 / α + β = 90
2 / it is enough to note that the triangles
KNP and MLQ (they have two heights of length "a") so they are isosceles
so |∡NMQ| =90 - α/2 and |∡MNP| =90 - β/2
γ=(α + β)/2=45
Two angles of a square with side protrude beyond a strip of width with parallel edges. The sides of the square intersect the edges of the strip at four points. Prove that the diagonals of the quadrilateral whose vertices are these points intersect at an angle of degrees.
Ok but answer is 10 not 0, so how solve this without wolfram.
https://hobbydocbox.com/Board_Games_and_Puzzles/77639583-Annual-ksf-meeting-november-protaras-cyprus.html
I've tried to find a trick solution.
Let $x, y, z$ be length of the side of a triangle such that $\sqrt{x} + \sqrt{y} + \sqrt{z} = 1.$
Prove $|x^{2} + y^{2} + z^{2} - 2\left( xy+yz+xz\right)| \le \frac{1}{27}$.
Let $ABC$ be a triangle with $\angle A= 60^{\circ},$ and $AD,BE$ are bisectors of $A,B$ respectively where $D\in BC, E\in AC.$ Find the measure of $B$ if $AB+BD=AE+BE.$
In a convex hexagon $ABCDEF$ exist a point $M$ such that $ABCM$ and $DEFM$ are parallelograms . Prove that exists a point $N$ such that $BCDN$ and $EFAN$ are also parallelograms.
There are 12 triangles (picture). We color each side of the triangle in red, green or blue. Among the $3^{24}$ possible colorings, how many have the property that every triangle has one edge of each color?
Adam has a circle of radius $1$ centered at the origin.
- First, he draws $6$ segments from the origin to the boundary of the circle, which splits the upper (positive $y$) semicircle into $7$ equal pieces.
- Next, starting from each point where a segment hit the circle, he draws an...