Anorexia Nervosa: This is the most likely diagnosis given the patient's rapid weight loss, obsession over her weight, refusal to eat with the family, and lack of appetite. The patient's body mass index (BMI) of 17.8, which is below the normal range, further supports this diagnosis. Anorexia nervosa is a psychological eating disorder characterized by an intense fear of gaining weight and a distorted body image, leading to restricted food intake and significant weight loss.

Bulimia Nervosa: Although the patient's symptoms primarily suggest anorexia nervosa, bulimia nervosa could also be considered, especially if the patient is purging or engaging in binge-eating behaviors that are not immediately apparent. However, the primary complaint of weight loss and lack of appetite leans more towards anorexia.
Depression: Depression can lead to significant changes in appetite and weight. While the patient's primary issue seems to be related to her weight and eating habits, underlying depression could be contributing to or exacerbating her condition.
Anxiety Disorder: Anxiety, especially if related to food or eating, could also lead to avoidance of meals and significant weight loss. However, the specific focus on weight and body image points more strongly towards an eating disorder like anorexia nervosa.

Diabetes Mellitus: Uncontrolled diabetes can lead to rapid weight loss due to the body's inability to properly use glucose. This would be a critical diagnosis not to miss, as untreated diabetes can lead to serious complications, including diabetic ketoacidosis.
Thyroid Disorders: Hyperthyroidism, in particular, can cause rapid weight loss, among other symptoms like increased appetite, anxiety, and heat intolerance. Missing this diagnosis could lead to untreated hyperthyroidism, which has significant health implications.
Gastrointestinal Disorders: Conditions like celiac disease, Crohn's disease, or ulcerative colitis can lead to malabsorption and weight loss. These conditions require specific management and could have serious consequences if left untreated.
Cancer: Although rare in adolescents, certain types of cancer (like lymphoma or leukemia) can present with weight loss. Given the potential severity of cancer, it's crucial not to miss this diagnosis, despite its lower likelihood.

Addison's Disease: This is a rare endocrine disorder where the adrenal glands do not produce enough cortisol and aldosterone. One of the symptoms can be weight loss, but it is accompanied by other specific signs like hyperpigmentation, hypotension, and electrolyte imbalances.
Pituitary or Hypothalamic Tumors: Tumors in these areas of the brain can affect hunger and satiety hormones, leading to weight loss. These conditions are rare and would likely be accompanied by other neurological symptoms.
Chronic Infections: Certain chronic infections, such as tuberculosis, can cause weight loss. However, they are typically accompanied by other symptoms like fever, night sweats, and cough.