How to construct a triangle from...?

Question

• Two medians and one height (nothing is for the same side!);
• Outer circle radius, one side and another side's height?

Using Compass-and-straightedge construction.

Answer 1

Two medians and one height 1

Given: $AE$, $BD$, $AF$.

1. Draw $AF$.
2. Construct perpendicular in $F$.
3. Draw circle around $A$ with radius $AE$.
4. Intersect 2. and 3. to obtain $E$.
5. Construct $G$ on $AE$ such that $AG:GE=2:1$.
6. Construct circle around $G$ with radius $\frac23\cdot BD$.
7. Intersect 2. and 6. to obtain $B$.
8. Measure distance $BD$ along $BG$ to find $D$.
9. Intersect $AD$ with 2. to find $C$.

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Two medians and one height 2

Given: $AE$, $BD$, $CF$.

1. Draw $BD$.
2. Construct $K$ as midpoint on $BD$.
3. Construct circle around $K$ through $B$ (and $D$).
4. Construct circle around $D$ with radius $\frac12\cdot CF$.
5. Intersect 3. and 4. to obtain $H$.
6. Construct $G$ on $BD$ such that $BG:GD=2:1$.
7. Construct circle around $G$ with radius $\frac23\cdot AE$.
8. Intersect 7. with $BH$ to obtain $A$.
9. Reflect $A$ in $D$ to obtain $C$.

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Outer circle radius, one side and another side's height

Given: Circumcircle radius, $AB$ and $AD$.

1. Draw circumcircle.
2. Construct side $AB$ as a segment of that circle.
3. Construct midpoint $E$ of $AB$.
4. Draw circle with center $E$ through $A$ (and $B$).
5. Draw circle around $A$ with radius $AD$.
6. Intersect 4. and 5. to get $D$.
7. Intersect $BD$ with circumcircle to get $C$.