ABCD is a parallelogram. The circle through A, B and C intersect CD (produced if necessary) at E. Prove that AE = AD.

Throught the mid-point M of the sides of a parallelogram ABCD, the line BM is drawn intersecting AC at L, and AD produced to E. Prove that EL = 2 BL .

In the adjacent figure ABCD is a parallelogram and E is the midpoint of the side BC. If DE and AB are produced to meet at F, show that AF = 2AB .

In the figure, ABCDE is a pentagon. A line through B parallel to AC meets DC produced at F. Show that (i) ar (ACB) = ar (ACF) (ii) ar (AEDF) = ar (ABCDE)

The diagonal AC of a parallelogram ABCD intersects DP at the point Q, where ‘P’ is any point on side AB. Prove that CQ xx PQ = QA xx QD .

In, ABCDE is a pentagon. A line through B parallel to AC meets DC produced at F. Show that (i)ar (ACB) = ar (ACF) (ii)ar (AEDF) = ar(ABCDE)

In the given figure, two circles with centres O and O' touch externally at a point A. A line through A is drawn to intersect these circles at B and C. Prove that the tangents at B and C are parallel.

The side AB of a parallelogram ABCD is produced to any point P. A line through A and parallel to CP meets CB produced at Q and then parallelogram PBQR is completed . Show that ar (ABCD) = ar (PBQR)

In the following figure, ABCD is parallelogram and BC is produced to a point Q such that AD = CQ. If AQ intersect AD = CQ. If AQ intersect DC at P, Show that ar (BPC) = ar (DPQ)

In figure, bar(AB) is a diameter of the circle, bar(CD) is a chord equal to the radius of the circle. bar(AC) and bar(BD) when extended intersect at a point E. Prove that angle AEB = 60^@ .

ABCD is a parallelogram. AC and BD are the diagonals intersect at O. P and Q are the points of tri section of the diagonal BD. Prove that CQ" ||" AP and also AC bisects PQ (see figure).