## Task 12 - V00912

A triangle is to be constructed for which the coordinates of the following points are given:

a) Footpoint $F$ of the height $h_{a}(-2 ;+2)$

b) Midpoint $D$ of the side $A B=c(+1 ;-3)$

c) Midpoint $M$ of the circumcircle $(+2 ;+1)$

Describe the construction! Measure the sides of the triangle to the nearest millimeter! ( $1 \mathrm{~cm} \cong$ 1 unit in the coordinate system)
## Construction:

(1) Draw the points $D, F, M$ in a coordinate system.

(2) The foot of the altitude $F$ lies on the Thales circle over $A B$, i.e., a circle with the center $D$ and the radius equal to $D F$. Draw this circle.

(3) The circumcenter $M$ lies on the perpendicular bisector of $A B$. Draw $D M$ and construct a perpendicular $s$ to it.

(4) This perpendicular $s$ intersects the Thales circle over $A B$ at the points $A$ and $B$. Construct $A$ and $B$.

(5) Connect $A$ with the foot of the altitude $F$ and construct a perpendicular to $A F$. The midpoint of the segment $B C$ lies on this perpendicular. Since $M$ lies on the perpendicular bisector of $B C$, the midpoint $E$ of the segment $B C$ is obtained by dropping a perpendicular from $M$ to the perpendicular through $F$. The foot of this perpendicular is $E$.

(6) The point $C$ is then the result of a point reflection of $B$ over $E$.

Triangle sides: $c=11.7 \mathrm{~cm}, a=12.3 \mathrm{~cm}, b=13.3 \mathrm{~cm}$.

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0855.jpg?height=797&width=788&top_left_y=247&top_left_x=631)