ABC is a triangle with circumcenter O, obtuse angle BAC and AB less than AC. M and N are the midpoints of BC and AO respectively. Let D be the intersection of MN with AC. If 2AD=(AB+AC), what is the measure of angle BAC ?

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120 To find the measure of angle BAC, we can use the given information about the triangle ABC and the relationship between the sides and angles of a triangle.

Since M is the midpoint of BC, we know that BM = MC. Similarly, since N is the midpoint of AO, AN = NO.

Now, let's analyze the given condition: 2AD = AB + AC.

From triangle AMC, we have AD = (AM + MC)/2, where AM = BM and MC = BC/2. Therefore, AD = (BM + BC/2)/2.

From triangle ABN, we have AB = AN + BN, where AN = NO and BN = (BO + ON)/2. Since BO = AO and NO = AN, we can rewrite it as AB = 2AN + BN.

Substituting the values of AB and AD into the given condition, we get:

2((BM + BC/2)/2) = (2AN + BN) + AC

Simplifying the equation, we have:

BM + BC/2 = AN + BN + AC

Since BM = MC and AN = NO, we can rewrite it as:

BC/2 = NO + BN + AC

Since N is the midpoint of AO, NO = AN = AO/2. Substituting this into the equation, we get:

BC/2 = AO/2 + BN + AC

Now, let's consider triangle ABC with obtuse angle BAC. In an obtuse triangle, the circumcenter O lies outside the triangle. Therefore, AC = OC + OA.

Substituting this into the equation, we have:

BC/2 = AO/2 + BN + OC + OA

Notice that AO/2 + BN + OC represents the length of AD. Therefore, the equation can be rewritten as:

BC/2 = AD + OA

Since the equation is 2AD = AB + AC, we can conclude that BC/2 = AD + OA = AD + AD.

This implies that BC = 2AD.

Since BC = 2AD, and AB + AC is equal to 2AD, it means that AB + AC = BC.

In a triangle, the sum of any two sides must be greater than the third side. Since AB + AC = BC in this case, it implies that the triangle is degenerate, or it doesn't exist.

Therefore, there is no valid measure for angle BAC in this scenario.