# Prove that points A, B, K and L lie on a circle $c$

In an acute-angled triangle ABC with height CD, K and L are orthogonal projections through D respectively on AC and BC. Prove that points A, B, K and L lie on a circle $c$.

I tried to prove that triangles ADK and DLB are of the same form, but without any luck.

• Hint: the circle will have diameter $AB$. Nov 3, 2015 at 20:41
• I'm not sure what you mean. If you are claiming that $KA=KB$ or $KA=LA$, then this isn't correct. Nov 3, 2015 at 20:49
• @Wojowu The only hit I've got is "All points of which the distance to two given points is a fixed ratio, lie on a circle."
– user270346
Nov 3, 2015 at 20:51
• Can you use the fact that the circumcenter of a right triangle is the midpoint of the hypotenuse? If so, consider circumcircles of $ABK$ and $ABL$ and prove these are equal. Nov 3, 2015 at 20:53
• @Wojowu I am not sure what you mean. How do you come to the conclusion that ABK and ABL are right triangles?
– user270346
Nov 3, 2015 at 20:58

Hint: try to prove the following equalities and conclude from there:

$$\angle KAB=\angle CDK=\angle CLK$$

First notice that the $\angle CDK=\angle CBA$ because $\angle BKD$ and $\angle CDB$ are both $\pi/2$. For the same reason $\angle CDL=\angle CAB$. From this follows that $$\angle LDK=\angle CDK+\angle CDL=\angle CBA+\angle CAB$$ therefore $$\angle LDK+\angle ACB=\angle CBA+\angle CAB+\angle ACB=\pi.$$ This implies that the points $C,L,D,K$ lie in the same circle. Hence $$\angle DKL= \angle DCL.$$ Now notice that $$\angle LKB+\angle CAB=\angle DKL+\pi/2+\angle CAB=\angle DCL+\pi/2+\angle CAB$$ and from the fact that $\angle DCB+\angle DBC=\pi/2$ we get $$\angle LKB+\angle CAB=\angle DCL+\angle DCB+\angle DBC+\angle CAB=\pi.$$ So $\angle LKB+\angle CAB=\pi$ and $A,L,K,B$ lie in the same circle.

$$\begin{array}{} B⟂AB & BE∩KD={E} \\ AF⟂AB & AF∩DL={F} \\ \end{array}$$

$$ΔCDL∼ΔADF$$

$$\begin{array}{} \frac{CD}{DL}=\frac{DF}{AF} & ⇒ & AF=\frac{DL·DF}{CD} \end{array}$$

$$ΔCAD∼ΔBDE$$

$$\begin{array}{} \frac{CD}{AD}=\frac{BD}{BE} & ⇒ & BE=\frac{AD·BD}{CD} \end{array}$$

$$ΔADF∼ΔBDL$$

$$\begin{array}{} \frac{AD}{DF}=\frac{DL}{BD} & ⇒ & AD·BD=DL·DF \end{array}$$

$$\begin{array}{} AF=BE & ⇒ AE=BF & (ΔABE≅ΔABF) \end{array}$$

(angle in a semicircle): right triangles:

$$\begin{array}{} ΔAKE, ΔABE & AE \text{(diameter)} & \overset{\Huge\frown}{AKBE} \\ ΔABF, ΔBLF & BF\text{(diameter)} & \overset{\Huge\frown}{BLAF} \end{array}$$

$$\begin{array}{} AE∩BF=M & \text{(Parallelogram Diagonals Theorem)} \end{array}$$

semicircles ($$\overset{\LARGE\frown}{AKE}$$ and $$\overset{\LARGE\frown}{BLF})$$ with congruent diameters ($$AE = BF$$) and common centers (midpoint $$M$$) are arcs contained in circle $$M$$ (circle $$AKLB$$)