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In $\triangle ABC$, the circumcenter and orthocenter are collinear with vertex $A$. Which of the following statements must be true?

(1) $\triangle ABC$ must be an isosceles triangle.

(2) $\triangle ABC$ must be an equilateral triangle.

(3) $\triangle ABC$ must be a right triangle.

(4) $\triangle ABC$ must be an isosceles right triangle.

Hints such as

"How should I draw this triangle?"

please

 Nov 10, 2019
 #1
avatar+139 
+3

Hint: 

The circumcenter is where the perpendicular bisectors meet

 

The orthocenter is where the altitudes intersect

 Nov 10, 2019
 #2
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Triangle ABC must be right, so the answer is (3).

 Nov 10, 2019
 #3
avatar+2505 
+1

do any of the other answer choices work?

CalculatorUser  Nov 10, 2019
 #4
avatar+105989 
0

If it is a genuine question you could at least present it properly.

Why have you left all those meaningless $ signs there.

To me this is an indication of laziness.

 Nov 10, 2019
 #5
avatar+105989 
+2

You want hints?

ok

Find out what a circumcentre and a orthocentre is.  Google is very helpful.

Now draw a triangle and roughly indicate where C and O.     I suggest you draw an acute angled, isosceles triangle to start with.

Now think about the line from BC ot the circumcentre C.

How will the triangle have to change to get this to be the same line as BC to the orthocentre O

This will indicate at least on correct answer.  

Now think about it (draw some more triangles)  to see if any of the othes needs to be true as well

 

This took me some time to work out and then to check. It wil take you and brain power too.

 

Please no one answer over me.

 Nov 10, 2019
 #6
avatar+105989 
0

Actually, CalculatorUser also gave you an excellent idea.

Draw the suggested triangles and see which one will make it true!

 

Thanks CalculatorUser.

 Nov 10, 2019

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