ABCD is a rectangle the line through C perpendicular to AC meets AB produced at X and AD produce at Y . Prove that DBXY are concyclic
To prove that the points D, B, X, and Y are concyclic (lie on the same circle), we can use the concept of the cyclic quadrilateral and its properties.
First, let's try to visualize the given information:
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B ______ X
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| |
| |
Y | |
|_________|
A C D
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From the given information, we know that ABCD is a rectangle, and the line passing through C, perpendicular to AC, meets AB produced at X and AD produced at Y.
Now, let's proceed with the proof step by step:
Step 1: Show that triangle ADY is similar to triangle CXB.
Since ABCD is a rectangle, angle ADC is a right angle. As the line through C is perpendicular to AC, angle ACY is also a right angle. Therefore, angle ADY and angle ACY are vertically opposite angles and are equal.
Similarly, as ADY is a right triangle, angle YAD is 90 degrees. As ABCD is a rectangle, angle ABC is also 90 degrees. Therefore, angle YAD and angle ABC are equal.
From the two equal angles, we can conclude that triangle ADY is similar to triangle CXB by the Angle-Angle similarity criterion.
Step 2: Show that XBYD is a cyclic quadrilateral.
Since triangle ADY is similar to triangle CXB, we can say that the corresponding angles of these triangles are equal.
From the similar triangles, we have:
∠ADY = ∠CXB -- (1)
∠AYD = ∠BCX -- (2)
Since ABCD is a rectangle, angle AYD is 90 degrees. Therefore, ∠BCX is also 90 degrees.
From equation (1), we have ∠ADY = ∠CXB = 90 degrees.
This implies that XBYD is a cyclic quadrilateral because the opposite angles of a quadrilateral inscribed in a circle are supplementary.
So, we have successfully proved that D, B, X, and Y are concyclic (lie on the same circle).
Therefore, DBXY are concyclic.