Prove: All triangles are isosceles triangles.
Proof. Given any ABC. Let ray r be the bisector of ACB. Let M be the midpoint of
AB. Let l be the line perpendicular to AB at M. Let D be the point of intersection of ray
r and line l. We consider two cases where D = M and D ≠ M.
Case 1. Assume D = M. We have r  CM and l  CM .
Then AMC  BMC , since l  AB at M.
Since M is the midpoint of AB, AM  MB.
Also, CM  CM .
Hence, by SAS AMC  BMC.
Thus, AC  BC and ABC is isosceles.
Case 2. Assume D  M. Then AMD  BMD, since l  AB at M.
Since M is the midpoint of AB, AM  MB.
Also, DM  DM .
Hence, by SAS AMD  BMD.
Thus, AD  BD.
Let E be the foot of the perpendicular line from D to BC. Let F be the foot of the
perpendicular line from D to AC.
Then CFD and CED are right angles.
Thus CFD  CED.
Since r  CD bisects ACB, FCD  ECD.
Also, CD  CD.
Hence, by AAS FCD  ECD.
Thus FD  ED and FC  EC.
Since AD  BD, FD  ED, and AFD and BED are right angles, by HL
AFD  BED.
Hence, AF  BE.
Since AF  BE and FC  EC , we have that AC = AF + FC = BE +EC = BC.
Hence, AC  BC.
Therefore, ABC is isosceles. □
Identify the error/s in the proof. Is the error a logic error? a bad assumption? Explain.
Directions for Wingeom to find the error in the proof that “Every triangle is
isosceles.”
The program opens with no work area window open. To start a work area, click on
Window on the toolbar and choose 2-dim.
To make a construction, from the main toolbar click on Btns and choose toolbar.
This opens a useful toolbar that makes moving between types of objects simpler.
1. Draw triangle ABC.
(a) click on segments in the toolbar
(b) plot the three vertices of the triangle by right-clicking the mouse at the desired
positions
(c) draw the segments by holding the left mouse button down and drag from
endpoint to endpoint
(d) to color the interior of the triangle: from the main toolbar click on Edit, then
choose Highlights Fill regions… , in the dialog box polygon enter the letters of
the triangle ABC and choose the Color, then click on fill
2. Bisect angle ACB.
(a) from the main toolbar click on Line, then choose Angles Bisect old…
(b) in the dialog box enter ACB
(c) change the letter D to Y by selecting text edit from the toolbar then right
click on D, type Y in the dialog box Note: This leaves D available for a name.
(d) hide point Y by selecting drag vertices, right click on the point Y
3. Construct the perpendicular bisector of segment AB.
(a) from the main toolbar click on Line, then choose Perpendiculars
Perp bisectors…
(b) in the dialog box enter AB
(c) change the name of the midpoint to M as in step 2c
(d) change the name of point E to Z and hide Z as in step 2c and step 2d
4. Plot the point of intersection of the angle bisector (ray CY) and the perpendicular
bisector (line MZ).
(a) from the main toolbar click on Point, then choose Intersection Line-line…
(b) in the dialog boxes enter CY and MZ, then click on mark
(c) this intersection point should be named D
5. Construct perpendiculars from D to the sides AC and BC of triangle ABC.
(a) from the main toolbar click on Line, then choose Perpendicular Altitudes
(b) in the dialog box enter the name of the line BC and point D, then click on draw
Repeat for AC and D.
6. Drag the vertices of the triangle to determine the problem with the proof.
(a) select drag vertices from the toolbar
(b) use the cursor and hold the left mouse button to drag a point